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Global well posedness for the semilinear edge-degenerate parabolic equations on singular manifolds

  • Yuxuan Chen EMAIL logo
Published/Copyright: November 28, 2023
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Advances in Nonlinear Analysis
From the journal Advances in Nonlinear Analysis Volume 12 Issue 1

Abstract

In this article, we study the long-time dynamical behavior of the solution for a class of semilinear edge-degenerate parabolic equations on manifolds with edge singularities. By introducing a family of potential well and compactness method, we reveal the dependence between the initial data and the long-time dynamical behavior of the solution. Specifically, we give the threshold condition for the initial data, which makes the solution exist globally or blowup in finite-time with subcritical, critical, and supercritical initial energy, respectively. Moreover, we also discussed the long-time behavior of the global solution, the estimate of blowup time, and blowup rate. Our results show that the relationship between the initial data and the long-time behavior of the solution can be revealed in the weighted Sobolev spaces for nonlinear parabolic equations on manifolds with edge singularities.

Keywords: global existence; finite-time blowup; nonlinear parabolic equation; singular manifolds
MSC 2010: 35K20; 35K55

1 Introduction

The focus of this article is to analyze the following initial boundary value problem for semilinear edge-degenerate parabolic equations on manifolds with edge singularities:

(1.1) u t − Δ E u = ∣ u ∣ p − 1 u , ( w , x , y ) ∈ int E , t > 0 ,

(1.2) u ( x , 0 ) = u 0 , ( w , x , y ) ∈ int E ,

(1.3) u ( x , t ) = 0 , ( w , x , y ) ∈ ∂ E , t ≥ 0 ,

where u 0 ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) , 1 < p < N + 2 N − 2 , N = 1 + n + n ′ ≥ 3 is the dimension of E , n , n ′ ∈ N . Here, E = [ 0 , 1 ) × X × Ω is regarded as a local model near the boundary of the stretched edge manifold (i.e., a manifold with edge singularity, see Section 2 for details). int E denotes the interior of E , the boundary of E is denoted by ∂ E = { 0 } × X × Ω , X is a closed set in R n , n ≥ 1 , Ω is an open domain in R n ′ , n ′ ≥ 1 , the dimension of E is N = 1 + n + n ′ ≥ 3 , and the coordinates ( w , x , y ) = ( w , x 1 , … , x n , y 1 , … , y n ′ ) ∈ E . The edge-Laplacian operator

Δ E = ∇ E 2 = ( w ∂ w ) 2 + ∂ x 1 2 + … + ∂ x n 2 + ( w ∂ y 1 ) 2 + … + ( w ∂ y n ′ ) 2

is an elliptic operator with edge degeneracy on the boundary ∂ E , where

∇ E = ( w ∂ w , ∂ x 1 , … , ∂ x n , w ∂ y 1 , … , w ∂ y n ′ )

denotes the gradient operator with edge degeneracy on the boundary ∂ E . As a description of the spatial derivatives, the degenerate elliptic operator Δ E appearing in (1.1) is regarded as the typical degenerate differential operators on a stretched edge manifold (cf. [28,33,35,37,51]).

The main features of this study originate from singular manifolds with twisted wedge-shaped, and problems with such geometric singularities often arise in differential geometry, porous medium theory, and fluid dynamics. For details about the concepts, symbols, and properties of the aforementioned stretched edge manifold E and the differential operators Δ E on it, please refer to Section 2 in this article. For the convenience of the readers, let us put aside the details and first outline the motivation and context of this article. Focusing on the dependence of the long-time dynamical behavior of solutions on initial data, we begin by presenting the results on the parabolic equations on the Euclidean domain, followed by an account of recent research on nonlinear parabolic equations related to (1.1)–(1.3) on manifolds with edge singularities. Based on these existing results, it is a mathematically and physically interesting problem whether certain conclusions regarding the dynamical behavior of solutions that have already been drawn on the Euclidean domain will hold in the singular domain. Furthermore, the difficulties posed by edge-degenerate operators Δ E for studying the dynamical behavior of the solution are also pointed out. To this end, we take initial data as a starting point to explore its influence on the dynamical behavior of the solution to Problems (1.1)–(1.3) and not only reveal the dependence of the initial data on various aspects of the dynamical properties of the solution but also give a comprehensive picture of the dynamical behavior of the solution under different initial data conditions in terms of energy, one by one.

There is an extensive literature related to the research on the well posedness of solutions to semilinear parabolic equations, which received a great deal of attention during the last few years around various perspectives on the relationship between the local well posedness [53], global existence [18], and finite-time blowup [14,54] of solutions and the corresponding geometric properties [27], spectral properties [23], nonlinear indicators [21], other factors, etc., in [43,47,57]. Of all the studies that used different perspectives and methods to deal with the influence of a factor on the dynamical behavior of solutions to the nonlinear parabolic equations, we are more interested in the effect of initial data on the well posedness of solutions. Among the many approaches that focus on initial data, the potential well method [44,48] proposed by Payne and Sattinger is one of the powerful tools to address it. Indeed, the main advantage of using the potential well theory to problem is that it allows us to discuss the existence of global solutions to nonlinear parabolic equations in case of non-positive definite energy. Meanwhile, it allows us to obtain both solutions that exist globally and blowup in finite-time realized by dividing initial data under different initial energy levels. Early analysis of the dynamical behavior of solutions by the potential well method usually located in the phase space over the Euclidean domain. For example, Gazzola and Weth [24] discussed the semilinear parabolic equations on a bounded smooth domain Ω ⊂ R n , i.e.,

(1.4) u t − Δ u = ∣ u ∣ p − 1 u .

Paying attention to the initial boundary value problem of equation (1.4) on Ω ⊂ R n , they not only comprehensively summarized the known results on the dynamical behavior of solutions partitioned by initial data within J ( u 0 ) ≤ d , but also used the comparison principle and variational methods to show the existence and nonexistence of solutions by the new idea that exploits the weak dissipativity of the semiflow inside (or outside) the Nehari manifold at high initial energy case J ( u 0 ) > d . In addition, there are many nonlinear parabolic equations with complex structures, which can be well treated within the framework of potential wells. Considering the parabolic equations with logarithmic nonlinearity, Chen and Xu [17], Chen and Tian [13] divided the initial data leading to the global existence and non-existence of solutions at low initial energy J ( u 0 ) < d and obtained the infinite-time blowup of solutions. And also for the nonlinear parabolic equations with more complex structures, such as nonlinear parabolic equations with singular potential [31], Kirchhoff diffusion equations with nonlocal term [10,19], systems of coupled parabolic equations [55], or even the system of coupling diffusion equations and wave equations [16,39], the dependence of the global well posedness of solutions on initial data can be well treated by the potential well method effectively and flexibly.

Recently, Chen and Liu [11] considered the asymptotic stability and blowup of solutions for a class of nonlinear edge-degenerate parabolic equations with singular potentials on wedge singular manifolds in form of:

(1.5) u t − Δ E u − ε V u = ∣ u ∣ p − 1 u ,

where 0 < ε < ( C * ) − 2 , and C * is the best Sobolev embedding constant. By introducing a family of potential wells, they proved the existence of global solutions that exponentially decay under the initial energy level J ( u 0 ) ≤ d as long as initial data satisfy I ( u 0 ) > 0 . Otherwise, if initial data located in the manifold given by I ( u 0 ) < 0 and J ( u 0 ) ≤ d , then the solution blows up in finite-time; moreover, the lower-bound of blowup time was also given in this situation. Subsequently, Xu et al. [56] studied the global existence and non-existence for a semilinear edge-degenerate parabolic equation with singular potential term:

(1.6) u t − Δ E u − ε V u = α u + ∣ u ∣ p − 1 u ,

in which ε and α are two constants satisfying 0 ≤ ε < ( C * ) − 2 and 0 < α < λ 1 ( 1 − ( ε C * ) 2 ) , where C * and λ 1 are the optimal constant in edge-type Hardy’s inequality and the principal eigenvalues of the eigenvalue problem − Δ E ψ = λ ψ , respectively. Applying the potential well method, they obtained some conditions with different energy levels such that the solution exists globally with exponential decay and blows up in finite-time, respectively. Moreover, the existence of ground-state solution to the stationary problem was also discussed. As a connection of solutions between the evolution problem of equation (1.6) and the stationary problem of equation (1.6), they showed that all global bounded solution converges to a stationary solution. Based on these results, they discussed the instability of the ground-state solution of equation (1.6), and obtained a sufficient condition for Problem (1.6) at the arbitrary initial energy such that the solution blows up in finite-time, with a lifespan of the blow-up solution. Meanwhile, several attempts have been made in the global analysis for solutions of nonlinear evolution equations under the influence of the domain that contains the singularity such as conical point, edge, and corner in the boundary. For instance, combining the existence of bounded imaginary powers for suitable closed extensions of the bilaplacian and singular analysis, Roidos and Schrohe [45,46] proved the short-time solvability of the Cahn-Hilliard equation and Allen-Cahn equation in L p -Mellin-Sobolev spaces and obtained the asymptotics of the solution near the conical points. Recently, Chen et al. [15] have obtained the global well posedness, finite-time blowup of solutions for semilinear parabolic equations with singular potential on manifolds with conical singularities at high initial energy levels using Nehari flow, and energy estimates in the Mellin-Sobolev space. For more topics on nonlinear partial differential equations on singular spaces, one can see [12,20] and references therein.

The main particularity of equation (1.5) is the presence of degenerate elliptic operator Δ E , where the operators are locally interpreted as pseudo-differential operators along the edge with cone operator-valued symbols that are parameterized by the points of the cotangent bundle of the edge [50]. In fact, the crucial way that the geometric singularities of the underlying manifolds presented in many models eventually enter into consideration is through particular forms of the operators, here like the operators Δ E considered in this article. Meanwhile, the exciting phenomena are connected with the geometric singularities, for instance, in mechanics, elasticity theory, and cracks in the medium described by hypersurfaces with a boundary [49]. More specifically, to allow accurate predictions of molecular properties in quantum chemistry, the particle models in electronic structure theory need to be developed [26]. It is crucial to know the existence and dynamical behavior of solutions of these models near coalescence points of particles as an initial boundary value problem corresponding to the models, which are typically treated as cones, edges, and corner singularities embedded in the space of electronic configurations [22]. The model equation (1.5) considered in singular geometric spaces always brings many difficulties in analysis. Some of these challenging issues are from the absence of the doubling property[1] [12]. The doubling property plays a crucial role in setting up several element inequalities such as Poincaré inequality [25]. Hence, Fan [20] constructed the edge Sobolev inequality and edge Poincaré’s inequality for the weighted Mellin-Sobolev spaces working on a manifold with edge singularities and then investigated the asymptotic stability of solutions for the initial boundary value problem of equation (1.5) on wedge singular manifolds in [20]. Due to the singular structures on the boundaries of the manifold, the differential topology of the manifold makes the classical derivatives fail and bring a considerable shock to classical analytical theory. One of the first to focus on such problems was Kondrat’ev and Oleinik [29], who proposed using the totally characteristic degeneracy operators and the Mellin transformation to quantitatively characterize the calculus of the derivatives near the singularity. Subsequently, much attention has been paid to studying the corresponding pseudo-differential operators on singular manifolds. Obviously, it is impossible here to give a complete list of references. Let us mention, in particular, the monograph and articles of Schulze [49,50], Melrose [36], Nistor [40], Monthubert and Nistor [38], Witt [52], etc. Further references on the edge operators and the edge calculus of singularity may be found in Mazzeo and Vertman [34], Ammann et al. [3], Nistor [41,42], and Mathai and Melrose [32], etc.

Inspired by the aforementioned work, it is interesting to consider the long-time behavior of the solution for the initial boundary value problem of equation (1.4) on a manifold with edge singularities. Indeed, when studying the qualitative properties of the solution on a wedge singular manifold, the difficulties posed by the singularities on the boundary of the manifold to the analysis need to be overcome as the domain is no longer a smooth Euclidean domain. Furthermore, the classical analysis theory is no longer true for singular manifolds. When we consider the effect of initial data on the long-time dynamical behavior of solutions in a functional space with degenerate measures, compared to the results obtained by Gazzola and Weth [24] on classical Euclidean domains, shall we obtain similar conclusions? In addition, since the problem in [56] is considered in the range of 0 < α < λ 1 ( 1 − ( ε C * ) 2 ) , the critical case α = 0 naturally attracts our attention. We hope to discuss the long-time dynamical behavior of the solution in the critical case α = 0 to reveal the special role of parameter α in describing the dynamics characteristics of the solution. Of course, Problem (1.4) with critical parameters α = 0 no longer has better compactness conditions than the problem in [56], so we strive to give a detailed description of the dynamical behavior of the solution, which is beneficial to observe the differences between them. Such a comparison is undoubtedly essential and interesting. We must construct a new technique for degenerate differential operators on singular manifolds in an appropriate functional space. Although the edge Sobolev inequality and the edge Poincaré inequality fit into weighted Sobolev spaces, they have been proposed in [12]. However, these tools are still not enough when we want to consider in more detail the long-time dynamical behavior of the solution. For example, when considering the local existence of the solution, does an edge-degenerate operator act as an infinitesimal generator of semigroup? When focusing on the dynamical behavior of solutions under high initial energy levels, we note that F. Gazzola uses the comparison principle from classical theory, yet what tools should be built to investigate this on singular manifolds where the boundary is no longer smooth? All these are crucial questions that must be addressed if we want to reveal the long-time dynamical behavior of the solution to Problems (1.1)–(1.3).

Our aim is to enrich and refine the understanding of the dependence between the dynamical behavior of the solution and the initial data to problems (1.1)–(1.3). To this end, we overcome the technical difficulties posed by the manifold with edge singularities to the analysis and reveal the inherent relationship between the initial data and the dynamical behavior of the solution, in detail:

(i) In order to obtain the local existence of the solution, we discuss the spectral properties satisfied by the edge-degenerate operator Δ E and verify the edge-degenerate operator that satisfies the conditions for the generation of strongly continuous semigroups. Then, we obtain the local existence and uniqueness of the solution to Problems (1.1)–(1.3) via the semigroup method.

(ii) Using the potential well method, we divide the initial energy level into four parts, namely, the negative initial energy level, the subcritical initial energy level, the critical initial energy level, and the supercritical initial energy level, and we discuss the dynamical behavior of the solution under each initial energy-level condition in turn. In order to discuss the global existence and nonexistence of the solution under supercritical initial energy level, we reconstruct the generalized comparison principle in weighted Sobolev spaces for the first time and discuss the steady-state problem of Problems (1.1)–(1.3) in aid of nonlinear semigroup theory and critical point theory. Then, we prove the global existence and finite-time blowup of the solution under supercritical initial energy using some conclusions obtained by the steady-state problem and the Nehari flow.

As a novelty work, compared with the results given in [24] and [56], we shall give in detail the dynamic behavior of the solution to gain a deeper understanding of the dynamic properties of the solution for this problem. Specifically, for initial negative energy, we give by constructing invariant sets concerning initial data not only conclusions on finite-time blowup of the solution but also estimate the upper-bounds on blowup time and lower-bounds on blowup rates. For subcritical initial energy, we partition the initial data by the Nehari functional in virtue of the potential well depth and achieve the invariant set with respect to initial data through the dissipative structure of the equation. Furthermore, we prove the global existence and finite-time blowup of the solution using the compactness method and the concave function method, respectively, and obtain asymptotic estimates of the global solution, upper-bound estimates of the blowup time, and lower-bound estimates of the blowup rate using some differential inequalities. We extend the aforementioned conclusions to critical initial energy using the idea of scale transformation. For supercritical initial energy, we start from the steady-state problem and introduce the potential energy levels and the Nehari flow to prove the existence and nonexistence of the solution at any positive initial energy levels, based on which a sharp-like threshold condition for classifying the initial data is given. In particular, for arbitrarily positive initial energy, we also present a sufficient condition for determining the finite-time blowup of solutions.

In addition, given that the estimates for the blowup time in the aforementioned theorems are all upper-bound estimate, we add an estimate for the lower-bound of the blowup time.

The outline of this article is as follows:

  1. In Section 2, we introduce some notations and propositions concerning the weighted Sobolev spaces on manifolds with edge singularities, and a family of potential wells with corresponding properties;

  2. In Section 3, we show our main results;

  3. In Section 4, we give the proof of local existence and uniqueness of the solution;

  4. From Sections 5 to 8, we prove the dynamic behavior of the solution with non-positive initial energy, subcritical initial energy, critical initial energy, and supercritical initial energy, respectively.

2 Preliminaries

2.1 Wedge Sobolev spaces

Manifolds with edges. A manifold W with edges Y is locally (near points of the edge) modeled on a wedge, i.e., a Cartesian product between a model cone and an open set in Euclidean space { cone } × edge [33,49,50]. Intuitively, the local models of W near every y ∈ Y are wedge X Δ × Ω for X Δ = ( R ¯ + × X ) ∕ ( { 0 } × X ) and a negiborhood Ω of y on Y . The edges Y are allowed to be a disjoint union of components Y = Y 1 ∪ Y 2 ∪ … ∪ Y m of different dimensions; for simplicity, we assume that all connected components of Y have the same dimension n ′ . In particular, the bases X ( y 1 ) , X ( y 2 ) , … , X ( y m ) of the cones for different y 1 , y 2 , … , y m ∈ Y are diffeomorphic; we hence simply talk about it as X .

Setting the open stretched cone with base X as

X ∧ ≔ R + × X ≅ X Δ \ { 0 } ,

where R + = { ω ∈ R ∣ ω > 0 } , we have a splitting of variables ( ω , x , y ) in X ∧ × Ω . Using the cylindrical coordinates in ( R 1 + n \ { 0 } ) × Y , the stretched wedge W to W is simply R ¯ + × X × Y , which is a manifold with smooth boundary { 0 } × X × Y . The identification with X × Y gives us a trivial X -bundle over the edge Y . It often suffices to focus on the case of 0 ≤ w < 1 . Therefore, we consider in this article the local model of the following finite wedge:

E ≔ ( [ 0 , 1 ) × X ) ∕ ( { 0 } × X ) × Y ⊂ X Δ × Y ,

where X is a closed compact C ∞ manifold of dimension n , and Y is a bounded subset in R q . Then the corresponding stretched wedge to E is

E ≔ [ 0 , 1 ) × X × Y ,

with boundary ∂ E = { 0 } × X × Y .

Totally, a manifold W with boundary V that contains Y is depicted by a chain as W ⊃ V ⊃ Y of subspaces, making W \ Y and V \ Y are C ∞ manifolds, and W \ Y with boundary V \ Y is a C ∞ manifold, where any point of Y has neighborhood in W , which is homeomorphic to a wedge; in addition, the transition maps represent a specific “geometric” compatibility between different singular charts (Figure 1). Manifolds with boundary that contains edge form a category if one denotes morphisms W → W ˜ in this way that they are carved locally near edges by differentiable maps R + × X × Ω → R + × X ˜ × Ω ˜ that extend to differentiable maps R × X × Ω → R × X ˜ × Ω ˜ , where corresponding open sets Ω ⊂ R n ′ and Ω ˜ ⊂ R n ˜ ′ and cone bases X and X ˜ , respectively). Moreover, if analytical objects are discussed from the points of view of invariance, such as differential operators or distribution spaces on W \ Y , it is convenient to specify the cocycle of transition mapping. The invariance properties may be decided by the choice of the cocycle (e.g., some details related to the asymptotic property near edge singularity). Based on that, we often suppose the transition diffeomorphisms R + × X × Ω → R + × X × Ω ˜ to be independent of the axial variable w ∈ R + for small w . For such spaces with edges, we will not lose information; for their description, we do not need to refer to a maximal atlas.

Figure 1 Manifolds with edge singularities.
Figure 1

Manifolds with edge singularities.

Definition 2.1

(Manifolds with edges) A manifold W with boundary and edge is a Hausdorff topological space that contains a subspace Y , consisting of the edges, such that:

  1. W \ Y is a (paracompact and locally compact) C ∞ manifold of dimension 1 + n + n ′ with boundary;

  2. Y is a C ∞ manifold of dimension n ′ ≥ 1 ;

  3. Every y ∈ Y has an open neighborhood U in W such that there is a homeomorphism χ : U → X ∧ × Ω (call a singular chart) for a closed compact C ∞ manifold X = X ( y ) with C ∞ boundary ∂ X , i.e., X is the base of the model cone X ∧ = ( R ¯ + × X ) \ ( { 0 } × X ) of the local wedge X ∧ × Ω with open edge Ω ⊂ R n ′ , and χ induces diffeomorphisms χ ′ : U ∩ Y → Ω and χ = χ ∣ U \ Y : U \ Y → R + × X × Ω ;

  4. if χ ˜ : U → X ∧ × Ω ˜ is another singular chart for U , the transition map χ ˜ ∘ χ − 1 : R + × X × Ω → R + × X × Ω ˜ is the restriction of a diffeomorphisms R × X × Ω → R × X × Ω ˜ to R + × X × Ω (the restriction of the latter diffeomorphisms to { 0 } × X × Ω then induces in the third component a diffeomorphisms Ω → Ω ˜ ).

Set W be a manifold with boundary that contains edge Y , and take an atlas on W . The charts on W \ Y are that denote on W \ Y by a C ∞ structure, while every chart near Y is supposed to be singular in the aforementioned meaning. Due to the property of the singular charts, we can explain the set W \ Y as a subspace of a space W , i.e., the open stretched wedges of form R + × X × Ω is modeled locally near Y , and our atlas deduces a cocycle of transition maps in this way in which W is invariantly interpreted as a manifold with boundary that contains a specific kind of corner. W is called the stretched manifold related to W . The gap between W with W \ Y is composed of a space W sing in which the singular charts cause a cocycle that denotes on Y by an X -bundle (i.e., a fire bundle on Y with fiber X ). Then, we take W reg ≔ W \ W sing , and it is only W \ Y , which can be explained as a C ∞ manifold with C ∞ boundary in the general meaning. Splittings of variables on W near W sing be locally defined by ( w , x , y ) ∈ R ¯ + × X × Ω , we frequently focus the observation on W reg , where w > 0 . A similar structure makes sense for V = ∂ W , which implies us to denote a stretched manifold V related to V , with the regular part V reg ≃ V \ Y and the singular part V sing that has the structure of a ∂ X -bundle on Y . According to the definition, there exist canonical maps

W → W and V → V ,

which, respectively, induce diffeomorphisms

W \ W sing → W \ Y and V \ V sing → V \ Y .

Edge structure. The edge structure V e on a smooth compact manifold X is uniquely related to the fibrosis of the boundary π : ∂ X → Y with fiber F , because the space of all smooth vector fields of X is tangent to the fibers of π at the boundary. It denotes a Lie algebra because of the closed under the ordinary bracket on vector fields. V e is also a finitely generated C ∞ ( X ) -module. We use local coordinates ( w , x , y ) , where w vanishes simply on the boundary, ( y 1 , … , y n ′ ) are the coordinates on Y lifted to ∂ X and then extended inward, which is easy to locally examine. In addition, ( x 1 , … , x n ) restricts to coordinates along each fiber at ∂ X . Up to now, all elements of the set

(2.1) V e = Span C ∞ { w ∂ w , w ∂ y 1 , … , w ∂ y n ′ , ∂ x l , … , ∂ x n }

belong to V e , which are independent, and generate it locally over C ∞ ( X ) . The maximality and independence of the set (2.1) imply that there is a bundle e T X naturally related to V e , making V e = C ∞ ( X , e T X ) , and which is provided with a map to the ordinary tangent bundle, ι e : e T X → T X , caused by the appropriate inclusion V e ↪ C ∞ ( X , T X ) ι e is an isomorphism over X ¯ as V e makes up of all smooth vector fields; however, it is neither injective nor surjective at the boundary. e T X is denoted by briefly approach that the vector fields (2.1) be a spanning set of sections, i.e., every V ∈ V e can be uniquely shown as:

V = a w ∂ w + ∑ b i w ∂ y i + ∑ c j ∂ x j , where a , b i , b j ∈ C ∞ ( X ¯ ) ,

where the coefficients a , b i , and c j evaluated at p ∈ X are the linear coordinates in e T p X . The dual to e T X , defined e T * X , is spanned locally by the 1-forms:

d w w , d y 1 w , … , d y n ′ w , … , d x 1 , … , d x n ,

which are singular as forms in the normal meaning but smooth as sections of e T * X .

Operators on manifolds with edges. The typical differential operators A on W in stretched coordinates ( w , x , y ) ∈ R + × X × Ω are edge-degenerate. In the following, we describe how a certain class of operators with uniformly controlled degeneracies may be used. These are called edge operators because they arise when nondegenerate elliptic operators are written in polar coordinates around an edge of a domain, or indeed around any distinguished submanifold. Let the ring of differential operators of edge-type denoted by Diff e ν ( W ) be the subset of the space Diff ν ( W \ Y ) [2] consisting of those that can be locally expressed as sums of products of elements of V e . Thus, in local coordinates, any A ∈ Diff ν ( W \ Y ) that are closed to Y in the variables ( ω , x , y ) ∈ X ∧ × Ω can be expressed as:

A = ω − ν ∑ j + ∣ α ∣ ≤ ν a j α ( ω , y ) ( − ω ∂ ω ) j ( ω ∂ y ) α = ω − ν A E ,

with coefficients a j α ∈ C ∞ ( R ¯ + × Ω , Diff ν − ( j + ∣ α ∣ ) ( X ) ) . The operator A E is called edge-Laplacian, which is regarded as a special case of the typical degenerate differential operators on a stretched edge-manifold. Edge degeneracy means that the differentiation with respect to y occurs in the combination t D y close to t = 0 . The weight factor “ t ” in front of the operator plays a more significant role in the edge theory than for the cone, cf. [12,45]. This type of degenerate operators is also of independent interest.

For convenience, we introduce the following hypotheses and symbols, which will be valid throughout this study, i.e., R + N ≔ R + × R n × R n ′ , where N = 1 + n + n ′ , n , n ′ ∈ N . And D ′ ( R + N ) denotes the dual space of C 0 ∞ ( R + N ) , which is the space of all distributions on R + N .

Definition 2.2

(The Lebesgue space L p ( R + N ) and L p γ ( R + N ) ) Let u ∈ D ′ ( R + N ) . We say that u ∈ L p ( R + N ) with 1 < p < ∞ , if

‖ u ‖ L p ( R + N ) = ∫ R + N w N ∣ u ( w , x , y ) ∣ p d σ 1 p < ∞ ,

where d σ = d w w d x 1 … d x n d y 1 w … d y n ′ w .

The weighted L p spaces with weight γ ∈ R are denoted by L p γ ( R + N ) , i.e., if u ∈ L p γ ( R + N ) , then ω − γ u ∈ L p ( R + N ) , which equipped with the following norm:

‖ u ‖ L p γ ( R + N ) = ∫ R + N w N ∣ w − γ u ( w , x , y ) ∣ p d σ 1 p < ∞ .

Definition 2.3

(The ℋ p m , γ ( R + N ) space) For m ∈ N and γ ∈ R , we denote the spaces

ℋ p m , γ ( R + N ) ≔ u ∈ D ′ ( R + N ) ∣ w N p − γ ( w ∂ w ) k ∂ x α ( w ∂ y ) β u ∈ L p ( R + N ) ,

for k ∈ N , multi-index α ∈ N n , β ∈ N q with k + ∣ α ∣ + ∣ β ∣ ≤ m , i.e., if u ∈ ℋ p m , γ ( R + N ) , then ( w ∂ w ) k ∂ x α ( w ∂ y ) β u ∈ L p ( R + N ) . Thus, ℋ p m , γ ( R + N ) is a Banach space with the following norm:

‖ u ‖ ℋ p m , γ ( R + N ) = ∑ k + ∣ α ∣ + ∣ β ∣ ≤ m ∫ R + N w N ∣ w − γ ( w ∂ w ) k ∂ x α ( w ∂ y ) β u ( w , x , y ) ∣ p d σ 1 p .

Moreover, the subspace ℋ p , 0 m , γ ( R + N ) of ℋ p m , γ ( R + N ) denotes the closure of C 0 ∞ ( R + N ) with respect to the ‖ ⋅ ‖ ℋ p m , γ ( R + N ) norm.

Now, we extend the weighted p -Sobolev spaces on X ∧ × Y , where X ∧ × Y = R + × X × Y is the open stretched wedge.

ℋ p m , γ ( X ∧ × Y ) ≔ { u ∈ D ′ ( X ∧ × Y ) ∣ w N p − γ ( w ∂ w ) k ∂ x α ( w ∂ y ) β u ∈ L p ( X ∧ × Y , d σ ) } ,

for k ∈ N , multi-index α ∈ N n , β ∈ N q with k + ∣ α ∣ + ∣ β ∣ ≤ m . Then, ℋ p m , γ ( X ∧ × Y ) is a Banach space with the norm:

‖ u ‖ ℋ p m , γ ( X ∧ × Y ) = ∑ k + ∣ α ∣ + ∣ β ∣ ≤ m ∫ X ∧ × Y w N ∣ w − γ ( w ∂ w ) k ∂ x α ( w ∂ y ) β u ( w , x , y ) ∣ p d σ 1 p .

The subspace ℋ p , 0 m , γ ( X ∧ × Y ) of ℋ p m , γ ( X ∧ × Y ) is defined as the closure of C 0 ∞ ( X ∧ × Y ) .

Thus, we have the following definition.

Definition 2.4

(The ℋ p m , γ ( E ) space). Let E be the stretched wedge to the finite wedge E , then ℋ p m , γ ( E ) for m ∈ N , γ ∈ R denotes the subset of all u ∈ W loc m , p ( int ( E ) ) such that

ℋ p m , γ ( E ) = { u ∈ W loc m , p ( int ( E ) ) ∣ ω u ∈ ℋ p , 0 m , γ ( X ∧ × Y ) }

for any cut-off function ω , supported by a collar neighborhood of [ 0 , 1 ] × ∂ E .

Moreover, the subspace ℋ p , 0 m , γ ( E ) of ℋ p , γ m , γ ( E ) is defined as follows:[3]

ℋ p , 0 m , γ ( E ) = [ ω ] ℋ p , 0 m , γ ( X ∧ × Y ) + [ 1 − ω ] W 0 m , p ( int ( E ) ) ,

where the classic Sobolev space W 0 m , p ( int ( E ) ) denotes the closure of C 0 ∞ ( int ( E ) ) in W m , p ( E ˜ ) for E ˜ , i.e., a closed compact C ∞ manifold of dimension N containing E as a submanifold with boundary.

Proposition 2.5

(cf. [12] Edge-type Sobolev inequality) Let 1 ≤ p < N , 1 p * = 1 p − 1 N , and γ ∈ R . Assuming ω ∈ R + , x = ( x 1 , … , x n ) ∈ R n and y = ( y 1 , … , y n ) ∈ R q , we have the following estimate:

‖ u ‖ L p * γ * ( R + N ) ≤ c 1 ‖ ( ω ∂ ω ) u ‖ L p γ ( R + N ) + ( c 1 + c 2 ) ∑ i = 1 n ‖ ∂ x i u ‖ L p γ ( R + N ) + ( c 1 + c 2 ) ∑ i = 1 n ‖ ( ω ∂ y i ) u ‖ L p γ ( R + N ) + c 3 ‖ u ‖ L p γ ( R + N ) ,

for all u ∈ C 0 ∞ ( R + N ) , where γ * = γ − 1 , c 1 = α N , c 2 = α N ( N − 1 ) ( N − p γ ) N − p 1 N , c 3 = 1 N ( N − 1 ) ( N − p γ ) N − p 1 N , and α = ( N − 1 ) p N − p . Moreover, if u ∈ ℋ p , 0 1 , γ ( R + N ) , we have

‖ u ‖ L p * γ * ( R + N ) ≤ c ‖ u ‖ ℋ p , 0 1 , γ ( R + N ) ,

where the constant c = c 1 + c 2 .

Proposition 2.6

(cf. [12] Edge-type Poincaré inequality) If u ∈ ℋ p , 0 1 , γ ( R + N ) , where 1 < p < + ∞ and γ ∈ R , then

‖ u ‖ L p γ ( E ) ≤ d E ‖ ∇ E u ‖ L p γ ( E ) ,

where d E is the diameter of E .

Proposition 2.7

(cf. [11]) For 1 < l < 2 * = 2 N N − 2 , the embedding ℋ 2 , 0 1 , N + 1 2 ( E ) ↪ ℋ l , 0 1 , N + 1 l ( E ) is compact.

Proposition 2.8

(cf. [11] Edge-type Hölder inequality) If u ∈ L p N + 1 p ( E ) , v ∈ L p ′ N + 1 p ′ ( E ) with p , p ′ ∈ ( 1 , ∞ ) and 1 p + 1 p ′ = 1 , then we have the following edge-type Hölder inequality:

∫ E w q ∣ u v ∣ d σ ≤ ∫ E w q ∣ u ∣ p 1 p ∫ E w q ∣ v ∣ p ′ d σ 1 p ′ .

For convenience, we denote

( u , v ) E = ∫ E w q u v d σ and ‖ u ‖ p ≔ ‖ u ‖ L p N + 1 p ( E ) = ∫ E w q ∣ u ∣ p d σ 1 p .

Proposition 2.9

(cf. [20] Variational principle for the principal eigenvalue)

  1. The first eigenvalue of operator − Δ E is given by:

    λ 1 ( E ) = min { ( − Δ E u , u ) E ∣ u ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) w i t h ‖ u ‖ 2 = 1 } .

  2. Furthermore, the aforementioned minimum is attained for a function ψ 1 , positive within E , that solves

    − Δ E ψ 1 = λ 1 ( E ) ψ 1 , ( ω , x , y ) ∈ E , ψ 1 = 0 , ( ω , x , y ) ∈ ∂ E ,

    has a discrete set of positive eigenvalues { λ k } k ≥ 1 , which can be ordered, after counting (finite) multiplicity, as 0 < λ 1 ≤ λ 2 ≤ λ 3 ≤ … ≤ λ k ≤ … , and λ k → ∞ as k → + ∞ . Also, the corresponding eigenfunctions { ψ k } k ≥ 1 constitute an orthonormal basis of the Hilbert space ℋ 2 , 0 1 , N + 1 2 ( E ) , and we have the inequality as follows:

    λ 1 ‖ u ‖ 2 2 ≤ ‖ ∇ E u ‖ 2 2 , ∀ u ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) .

  3. Finally, if u ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) is any weak solution of

    − Δ E u = λ 1 ( E ) u , ( ω , x , y ) ∈ E , u = 0 , ( ω , x , y ) ∈ ∂ E ,

    then u is a multiple of ψ 1 .

Lemma 2.10

[30] Suppose that a positive, twice-differentiable function ψ ( t ) satisfies the inequality:

ψ ″ ( t ) ψ ( t ) − ( 1 + θ ) ( ψ ′ ( t ) ) 2 ≥ 0 , t > 0 ,

where θ > 0 is a constant. If ψ ( 0 ) > 0 and ψ ′ ( 0 ) > 0 , then there exists 0 < t 1 ≤ ψ ( 0 ) θ ψ ′ ( 0 ) such that ψ ( t ) tends to infinity as t → t 1 .

2.2 Potential wells

In this section, we introduce the potential well for Problems (1.1)–(1.3) and show a series of properties of the potential well, which are useful in the proof of our results. Let

(2.2) J ( u ) ≔ 1 2 ‖ ∇ E u ‖ 2 2 − 1 p + 1 ‖ u ‖ p + 1 p + 1 , for all u ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) ,

and

(2.3) I δ ( u ) ≔ δ ‖ ∇ E u ‖ 2 2 − ‖ u ‖ p + 1 p + 1 , for all u ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) , δ > 0 .

Since ℋ 2 , 0 1 , N + 1 2 ( E ) ↪ L p N + 1 p ( E ) , 1 ≤ p < N + 2 N − 2 , and Proposition 2.5, we know that the functionals J ( ⋅ ) , I δ ( ⋅ ) ∈ C 1 ( ℋ 2 , 0 1 , N + 1 2 ( E ) , R ) are well defined. In view of [4], J ( u ) satisfies the Palais-Smale condition and the mountain-pass level d 1 may be characterized as:

(2.4) d 1 = min u ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) \ { 0 } max λ ≥ 0 J ( λ u ) = inf u ∈ N 1 J ( u ) ,

where the Nehari manifold

(2.5) N 1 = { u ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) ∣ I 1 ( u ) = 0 , ‖ ∇ E u ‖ 2 ≠ 0 } .

Similarly, one can define the following depth of a family of potential wells:

d δ = inf u ∈ N δ J ( u ) ,

where N δ = { u ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) ∣ I δ ( u ) = 0 , ‖ ∇ E u ‖ 2 ≠ 0 } . Then, we give the following potential well:

(2.6) N δ + = { u ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) ∣ I δ ( u ) > 0 } ∪ { 0 } ,

and the outside set of the corresponding potential well

(2.7) N δ − = { u ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) ∣ I δ ( u ) < 0 } .

The index δ will be always omitted when equal to 1, i.e.,

I ( u ) = I 1 ( u ) , d = d 1 , N = N 1 , N + = N 1 + , N − = N 1 − .

Next, we show some properties for the aforementioned functionals, which also shows in [11] by a simple modification.

Lemma 2.11

Let u ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) \ { 0 } , considering the real-valued function J ( λ u ) and I ( λ u ) , λ ∈ [ 0 , ∞ ) , we have

  1. lim λ → 0 J ( λ u ) = 0 , and lim λ → + ∞ J ( λ u ) = − ∞ ;

  2. There is a unique λ * = λ * ( u ) > 0 , such that d d λ J ( λ * u ) = 0 ;

  3. J ( λ u ) is strictly increasing on 0 ≤ λ < λ * , strictly decreasing on λ > λ * , and takes the maximum at λ = λ * ;

  4. I ( λ u ) > 0 for 0 ≤ λ < λ * , I ( λ u ) < 0 for λ > λ * , and I ( λ * u ) = 0 .

Let S be the optimal embedding constant for ℋ 2 , 0 1 , N + 1 2 ( E ) ↪ L p + 1 N + 1 p + 1 ( E ) , i.e.,

S = inf u ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) \ { 0 } ‖ ∇ E u ‖ 2 ‖ u ‖ p + 1 .

Lemma 2.12

Assume that u ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) \ { 0 } and r ( δ ) ≔ δ S p + 1 1 p − 1 .

  1. If 0 < ‖ ∇ E u ‖ 2 < r ( δ ) , then I δ ( u ) > 0 ;

  2. If I δ ( u ) < 0 , then ‖ ∇ E u ‖ 2 > r ( δ ) ;

  3. If I δ ( u ) = 0 , then ‖ ∇ E u ‖ 2 ≥ r ( δ ) ;

  4. If I δ ( u ) = 0 , then J ( u ) > 0 , f o r 0 < δ < p + 1 2 ; = 0 , f o r δ = p + 1 2 ; < 0 , f o r δ > p + 1 2 .

Additionally, we show that several properties of d δ depend on δ as follows.

Lemma 2.13

(Characterization of d δ ) The potential well depth d δ satisfies

  1. d = d 1 = p − 1 2 ( p + 1 ) S − 2 ( p + 1 ) p − 1 ;

  2. d δ = δ 2 p − 1 2 ( p + 1 ) p − 1 1 2 − δ p + 1 d ≥ 1 2 − δ p + 1 r 2 ( δ ) , for all 0 < δ < p + 1 2 ;

  3. lim δ → 0 d δ = 0 , d p + 1 2 = 0 and d δ < 0 for δ > p + 1 2 ;

  4. The derivative of d δ is denoted as d δ ′ and satisfies d δ ′ > 0 , o n 0 < δ ≤ 1 ; = 0 , o n δ = 1 ; < 0 , o n 1 ≤ δ .

In addition, we also define

λ κ = inf { ‖ u ‖ 2 2 ∣ u ∈ N , J ( u ) < κ } and Λ κ = sup { ‖ u ‖ 2 2 ∣ u ∈ N , J ( u ) < κ }

for all κ > d . Obviously, we obtain the monotonicity properties:

κ ↦ λ κ is nonincreasing and κ ↦ Λ κ is nondecreasing .

As all parameters in (1.1) are fixed, we call that a function u = u ( t , z ) is a weak solution of Problems (1.1)–(1.3) corresponding to the initial data u 0 ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) on [ 0 , T ] × E , if it satisfies

  1. u ∈ L ∞ ( 0 , T ; ℋ 2 , 0 1 , N + 1 2 ( E ) ) and u t ∈ L 2 ( 0 , T ; L 2 N + 1 2 ( E ) ) ;

  2. u ( 0 ) = u 0 ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) ;

  3. For any v ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) , the identity

    ∫ E w q u t v d σ + ∫ E w q ∇ E u ∇ E v d σ = ∫ E w q ∣ u ∣ p − 1 u v d σ

    holds for a.e. t ∈ [ 0 , T ) .

And, the maximal existence time of the solution u is denoted by:

T max ( u 0 ) = sup { T > 0 : u = u ( t ) exists on [ 0 , T ] } .

Thus, we introduce the following sets related to the solution of Problems (1.1)–(1.3),

ℬ = u 0 ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) ∣ T max ( u 0 ) < ∞ , G = u 0 ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) ∣ T max ( u 0 ) = ∞

and the subsets

G 0 = u 0 ∈ G ∣ u ( t ) → 0 in ℋ 2 , 0 1 , N + 1 2 ( E ) as t → ∞ .

3 Main results

Theorem 3.1

(Local solution) Suppose that u 0 ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) , then there exists T > 0 such that Problems (1.1)–(1.3)possess a unique weak solution u on [ 0 , T ] × E , which satisfies

u ∈ C ( [ 0 , T ] , ℋ 2 , 0 1 , N + 1 2 ( E ) ) ∩ C ( ( 0 , T ] , ℋ 2 1 + ε , N + 1 2 ( E ) ) , t θ 2 ‖ u ( t ) ‖ ℋ 2 1 + θ , N + 1 2 ( E ) ⟶ t → 0 + 0 , 0 < θ < ε ,

for some ε : 0 < ε < 1 . Moreover, if T max = T max ( u 0 ) < ∞ , then lim t → T max ‖ ∇ E u ( t ) ‖ 2 = ∞ .

Theorem 3.2

(Blow up with J ( u 0 ) ≤ 0 ) Assume that u 0 ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) with J ( u 0 ) ≤ 0 , then

  1. There exists a bounded region

    (3.1) U r * = u ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) ∣ ‖ ∇ E u ‖ 2 < r *

    for some r * > 0 , such that u ( t ) ∉ U r * for all t ∈ [ 0 , T max ) .

  2. u blows up in finite-time with L 2 N + 1 2 ( E ) norm and T max is estimated as follows:

    T max ≤ 2 − p + 1 p − 1 ( p + 1 ) ( 1 − p ) 3 − p p − 1 ∣ E ∣ p − 1 2 ‖ u 0 ‖ 2 1 − p , when J ( u 0 ) = 0 ; − ( J ( u 0 ) ) − 1 ( p − 1 ) − 2 ‖ u 0 ‖ 2 2 , when J ( u 0 ) < 0 ,

    where ∣ E ∣ denotes the volume of the domain E . The blowup rate is estimated by;

    ‖ u ( t ) ‖ 2 ≤ ( p − 1 ) 4 1 − p ∣ E ∣ 2 ( p + 1 ) 2 1 − p 1 − 2 p + 1 p + 1 p − 1 2 1 − p ( T max − t ) 2 1 − p , when J ( u 0 ) = 0 ; ( 1 − p 2 ) J ( u 0 ) ‖ u 0 ‖ 2 p + 1 1 1 − p ( T max − t ) − 1 p − 1 , when J ( u 0 ) < 0 ;

    Moreover, u grows exponentially with L p + 1 N + 1 p + 1 ( E ) norm for t ∈ [ 0 , T max ) .

Theorem 3.3

(Sharp threshold for 0 < J ( u 0 ) < d ) Let u 0 ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) with 0 < J ( u 0 ) < d , and δ 1 < δ 2 are two roots of d δ = J ( u 0 ) .

  1. If u 0 ∈ N + , then u 0 ∈ G 0 and u ( t ) ∈ N δ + for all t ≥ 0 and δ ∈ ( δ 1 , δ 2 ) . Moreover, the global weak solution satisfies

    (3.2) ‖ ∇ E u ( t ) ‖ 2 2 + ∫ 0 t ‖ u t ( τ ) ‖ 2 2 d τ < 2 ( p + 1 ) ( p − 1 ) − 1 d , t ≥ 0 ,

    and the decay estimates

    (3.3) ‖ u ( t ) ‖ 2 2 ≤ ‖ u 0 ‖ 2 2 e − 2 λ 1 t 1 − S p + 1 2 ( p + 1 ) p − 1 J ( u 0 ) p − 1 2 , t ≥ 0 .

  2. If u 0 ∈ N − , then u 0 ∈ ℬ and u ( t ) ∈ N δ − for all t ∈ [ 0 , T max ) and δ ∈ ( δ 1 , δ 2 ) . Moreover, the upper-bound of blowup time can be estimated by:

    (3.4) T max ≤ ( p + 1 ) ∣ E ∣ p − 1 2 ( p − 1 ) − 2 1 − ( p + 1 ) 1 2 − J ( u 0 ) S 2 ( p + 1 ) p − 1 − p + 1 p − 1 − 1 ‖ u 0 ‖ 2 1 − p ,

    and for all t ∈ [ 0 , T max ) , the upper-bound of blowup rate can be estimated by:

    (3.5) ‖ u ( t ) ‖ 2 2 < C p 1 − ( p + 1 ) 1 2 − J ( u 0 ) S 2 ( p + 1 ) p − 1 − p + 1 p − 1 2 1 − p ( T max − t ) − 2 p − 1 ,

    where C p = 2 − 1 ( p − 1 ) 4 1 − p ∣ E ∣ ( p + 1 ) 2 p − 1 . Finally, we also conclude that the u increases exponentially in L p + 1 N + 1 p + 1 ( E ) norm for t ∈ [ 0 , T max ) .

Theorem 3.4

(Vacuum isolating phenomena of the solution) Suppose δ 1 < δ 2 are the two roots of d δ = e ∈ ( 0 , d ) . Then, there is a vacuum region

U e = ⋃ δ 1 < δ < δ 2 N δ = u ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) ∣ S − p + 1 p − 1 δ 1 1 p − 1 < ‖ ∇ E u ‖ 2 < S − p + 1 p − 1 δ 2 1 p − 1 ,

such that u ( t ) ∉ U e , t ∈ [ 0 , T max ) holds for all weak solutions u of Problems (1.1)–(1.3) with J ( u 0 ) ≤ e .

Theorem 3.5

(Sharp threshold for J ( u 0 ) = d ) Let u 0 ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) with J ( u 0 ) = d ,

  1. If u 0 ∈ N + , then u 0 ∈ G 0 . Moreover, the decay estimates

    ‖ ∇ E u ‖ 2 2 ≤ C e − t C 1 a n d J ( u ( t ) ) ≤ C 2 t − 1

    hold for t > 0 , where C , C 1 , and C 2 are constants.

  2. If u 0 ∈ N − , then u 0 ∈ ℬ . Moreover, for t > 0 , the upper-bound of blowup time can be estimated by:

    T max ≤ ( p + 1 ) ∣ E ∣ p − 1 2 ( p − 1 ) − 2 1 − ( p + 1 ) 1 2 − d S 2 ( p + 1 ) p − 1 − p + 1 p − 1 − 1 ‖ u 0 ‖ 2 1 − p ,

    and the upper-bound of blowup rate can be estimated by:

    ‖ u ( t ) ‖ 2 2 < ( p − 1 ) 4 1 − p ∣ E ∣ 2 ( p + 1 ) 2 1 − p 1 − ( p + 1 ) 1 2 − d S 2 ( p + 1 ) p − 1 − p + 1 p − 1 2 1 − p ( T max − t ) 2 1 − p .

    Finally, we also conclude that the u increases exponentially in L p + 1 N + 1 p + 1 ( E ) -norm for t ∈ [ 0 , T max ) .

Theorem 3.6

(Global existence and nonexistence with J ( u 0 ) > 0 ) Let u 0 ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) . For every M > 0 , we conclude that

  1. There exist u 0 , v 0 ∈ N + ∩ C 0 1 ( E ) satisfying u 0 ≥ 0 , v 0 ≥ 0 a.e. in E , J ( u 0 ) ≥ M , J ( v 0 ) ≥ M , and

    u 0 − v 0 = 2 ϕ M ,

    where C 0 1 ( E ) = C 1 ( E ¯ ) ∩ ℋ 2 , 0 1 , N + 1 2 ( E ) , the positive function ϕ M ∈ C 0 1 ( E ) satisfies ‖ ∇ E ϕ M ‖ 2 ≥ M a n d ‖ ϕ M ‖ ∞ ≤ ε for a sufficiently small ε > 0 . However, u 0 ∈ ℬ and v 0 ∈ G 0 ;

  2. There exists u 0 ∈ N − satisfying J ( u 0 ) ≥ M such that u 0 ∈ ℬ . In particular, if initial datum satisfies

    (3.6) ‖ u 0 ‖ 2 2 > 2 λ 1 − 1 ( p + 1 ) ( p − 1 ) − 1 J ( u 0 ) ,

    then u 0 ∈ ℬ , and the corresponding upper-bound of blowup time is estimated by:

    T max ≤ 8 ( p + 1 ) ‖ u 0 ‖ 2 2 λ 1 ( p − 1 ) 2 ‖ u 0 ‖ 2 2 − 2 ( p 2 − 1 ) J ( u 0 ) .

Theorem 3.7

(Sharp-like threshold for solutions with J ( u 0 ) > d ) Let u 0 ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) and J ( u 0 ) > d .

  1. If u 0 ∈ N + with ‖ u 0 ‖ 2 2 ≤ λ J ( u 0 ) , then u 0 ∈ G 0 ;

  2. If u 0 ∈ N − with ‖ u 0 ‖ 2 2 ≥ Λ J ( u 0 ) , then u 0 ∈ ℬ .

Theorem 3.8

(Lower bound of blowup time and blowup rate) Let u 0 ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) . Assume that u be the finite-time blowup solutions for Problems (1.1)–(1.3), then a lower-bound of blowup time to the solution is given by:

T max ≥ 36 λ 1 ( β − 1 ) 2 β 3 ( p − 1 ) ∣ E ∣ 1 − 3 ( p − 1 ) β ‖ u 0 ‖ β 3 ( p − 1 ) ,

where the odd number β fulfills β > 3 ( p − 1 ) . Moreover, the lower-bound of the blowup rate is estimated by:

‖ u ( t ) ‖ β ≥ β 4 ∣ E ∣ 1 − 3 ( p − 1 ) β 36 ( β − 1 ) 2 λ 2 − 1 3 ( p − 1 ) ⋅ ( T max − t ) − 1 3 ( p − 1 ) .

4 Local existence of solution

In this section, we prove the local existence and the comparison principle of weak solutions.

Proof of Theorem 3.1

Indeed, the stretched edge manifold E has proven to be the uniformly regular Riemannian manifolds by Amann [1,2]. Hence, according to the results in [8], we know that the operator ℒ = Δ E can be seen as an unbounded operator in V 0 = L 2 N + 1 2 ( E ) with domain V 1 = ℋ 2 2 , N + 1 2 ( E ) ∩ ℋ 2 , 0 1 , N + 1 2 ( E ) . By [7], the operator ℒ , which as a ℛ -sectorial operators on singular manifolds [8] is resolvent positive, is the infinitesimal generator of the analytic semigroup e ℒ t . The scale of fractional power space { V α } α ∈ R associated with ℒ satisfies[4]

V α ↪ ℋ 2 2 α , N + 1 2 ( E ) , V − α = V α ′ , α ≥ 0 , V 0 = L 2 N + 1 2 ( E ) , V 1 2 = ℋ 2 , 0 1 , N + 1 2 ( E ) , V 1 = ℋ 2 2 , N + 1 2 ( E ) ∩ ℋ 2 , 0 1 , N + 1 2 ( E ) .

Moreover, the realization ℒ α : D ( ℒ α ) = V α + 1 ⊂ V α → V α is an isometry. Taking X α = V α − 1 2 for all α ∈ R , we have that A = ℒ − 1 2 : X 1 ⊂ X 0 → X 0 is an isometry. With these notations, Problems (1.1)–(1.3) are reduced into the following abstract equation:

(4.1) u t = A u + Φ ( u ) , u ( 0 ) = u 0 ∈ X 1 ,

where Φ ( u ) = ∣ u ∣ p − 1 u is an ε 2 ( p + 1 ) -regular map relative to the pair ( X 1 , X 0 ) . Indeed, with the fact 1 ≤ p < N + 2 N − 2 and Proposition 2.5, we know that ℋ 2 1 − ε , N + 1 2 ( E ) ↪ L p + 1 N + 1 p + 1 ( E ) for some ε : 0 < ε ≪ 1 . Thus, we have

L p + 1 p p ( N + 1 ) p + 1 ( E ) = L p + 1 N + 1 p + 1 ( E ) ′ ↪ ℋ 2 ε − 1 , N + 1 2 ( E ) ↪ V ε − 1 2 = X ε 2 ,

and

X 1 + ε 2 ( p + 1 ) = V 1 2 + ε 2 ( p + 1 ) ↪ ℋ 2 1 + ε p + 1 , N + 1 2 ( E ) ↪ L p + 1 N + 1 p + 1 ( E ) .

A simple calculation and Proposition 2.8 show that

‖ Φ ( u ) − Φ ( v ) ‖ X ε 2 ≤ C ‖ Φ ( u ) − Φ ( v ) ‖ p + 1 p ≤ C p ∫ E w q ( ( ∣ u ∣ p − 1 + ∣ v ∣ p − 1 ) ∣ u − v ∣ ) p + 1 p d σ p p + 1 ≤ C p ∫ E w q ( ∣ u ∣ p − 1 + ∣ v ∣ p − 1 ) p + 1 p ⋅ p p − 1 d σ p − 1 p ⋅ p p + 1 ⋅ ∫ E w q ∣ u − v ∣ p + 1 p ⋅ p d σ 1 p ⋅ p p + 1 ≤ C p ( ‖ p ‖ p + 1 p − 1 + ‖ v ‖ p + 1 p − 1 ) ‖ u − v ‖ p + 1 ≤ C p ‖ p ‖ X 1 + ε 2 ( p + 1 ) p − 1 + ‖ v ‖ X 1 + ε 2 ( p + 1 ) p − 1 ‖ u − v ‖ X 1 + ε 2 ( p + 1 ) ,

for all u , v ∈ X 1 + ε 2 ( p + 1 ) , i.e., Φ is an ε 2 ( p + 1 ) -regular map relative to the pair ( X 1 , X 0 ) . Hence, it follows from Corollary 1 in [5] that, for each u 0 ∈ X 1 , there exists a T > 0 , which only depends on ‖ ∇ E u 0 ‖ 2 , such that Problem (4.1) possesses a unique solution u ∈ C ( [ 0 , T ] ; X 1 ) ∩ C ( ( 0 , T ] ; X 1 + ε 2 ) satisfying

u ( t ) = e A t u 0 + ∫ 0 t e A ( t − s ) Φ ( u ( s ) ) d s .

Thus, we show the local existence and uniqueness of weak solution of Problems (1.1)–(1.3), which satisfy

u ∈ C ( [ 0 , T ] , ℋ 2 , 0 1 , N + 1 2 ( E ) ) ∩ C ( ( 0 , T ] , ℋ 2 1 + ε , N + 1 2 ( E ) ) , t θ ‖ u ( t ) ‖ ℋ 2 1 + 2 θ , N + 1 2 ( E ) ⟶ t → 0 + 0 , 0 < θ < ε 2 .

Moreover, since the local existence T only depends on the norms of the initial data. Therefore, using the similar idea as shown in [9], the solution can be continued as long as ‖ ∇ E u ‖ 2 remains bounded. Hence, if T max = T max ( u 0 ) < ∞ , we have

□ lim t → T max ‖ ∇ E u ( t ) ‖ 2 = ∞ .

Remark 4.1

Let u be the weak solution corresponding to u 0 ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) . If we differentiate the map t ↦ J ( u ( u ) ) with respect to t and use (1.1), we obtain

(4.2) d d t J ( u ( t ) ) = − ‖ u t ( t ) ‖ 2 2 , t ∈ [ 0 , T max ) .

Integrating (4.2) over ( 0 , t ) , it follows from that

(4.3) ∫ 0 t ‖ u t ( τ ) ‖ 2 2 d τ + J ( u ) = J ( u 0 ) , t ∈ [ 0 , T max ) .

Moreover, multiplying (1.1) by u and then integrating it on E show

(4.4) d d t ‖ u ( t ) ‖ 2 2 = − 2 I ( u ( t ) ) , t ∈ [ 0 , T max ) .

Corollary 4.2

(Comparison principle) Suppose that u 0 , v 0 ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) such that u 0 ≥ v 0 a.e. in E , then the solution u ( t ) ≥ v ( t ) a.e. in E for all t ≥ 0 . Moreover, if u 0 ≢ v 0 , then for t > 0 , we attain

(4.5) u ( t ) > v ( t ) a.e. i n E .

Proof

Obviously, the mapping Φ denoted in (4.1) is increasing. Hence, using Theorem A.10 in [6], one can show the conclusions of Corollary 4.2 immediately.□

5 Subcritical initial energy

In this section, we prove Theorems 3.3 and 3.4. To do this end, we first show some lemmas.

Lemma 5.1

Let u ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) with 0 < J ( u ) < d , and δ 1 < δ 2 be the two roots of d δ = J ( u ) . Then, the sign of I δ ( u ) does not change for all δ 1 < δ < δ 2 .

Proof

Arguing by contradiction, since I δ ( u ) is an increasing continuous function with respect to δ , we assume that there exists a δ ′ ∈ ( δ 1 , δ 2 ) such that I δ ′ ( u ) = 0 . It follows from the fact J ( u ) > 0 that ‖ ∇ E u ‖ 2 ≠ 0 , which implies that u ∈ N δ ′ . Thus, we obtain

(5.1) d δ ′ = inf φ ∈ N δ ′ J ( φ ) ≤ J ( u ) = d δ 1 = d δ 2 .

Moreover, from Lemma 2.13, we have d δ 1 = d δ 2 < d δ ′ , which contradicts (5.1).□

Lemma 5.2

Let u 0 ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) \ { 0 } and δ 1 < δ 2 be two roots of d δ = l , 0 < l < d , then

  1. All weak solutions of (1.1)–(1.3) with J ( u 0 ) = l belong to N δ + for δ 1 < δ < δ 2 , provided that u 0 ∈ N + ;

  2. All weak solutions of (1.1)–(1.3) with J ( u 0 ) = l belong to N δ − for δ 1 < δ < δ 2 , provided that u 0 ∈ N − .

Proof

(i) Let u be a solution of (1.1)–(1.3) with u 0 ∈ N + and J ( u 0 ) = l and T max = T max ( u 0 ) . From Lemma 2.13, 1 ∈ ( δ 1 , δ 2 ) . Thus, by Lemma 5.1 and the fact that

I 1 ( u 0 ) = I ( u 0 ) > 0 , 0 < J ( u 0 ) = l < d ,

one can infer that I δ ( u ) > 0 for all δ ∈ ( δ 1 , δ 2 ) , i.e., u 0 ∈ N δ + for all δ ∈ ( δ 1 , δ 2 ) .

Next, we claim that u ( t ) ∈ N δ + for any δ ∈ ( δ 1 , δ 2 ) and t ∈ ( 0 , T max ) . Arguing by contradiction, since u ∈ C ( [ 0 , T ] ; ℋ 2 , 0 1 , N + 1 2 ( E ) ) , we assume that there exist a t 0 ∈ ( 0 , T max ) and a δ 0 ∈ ( δ 1 , δ 2 ) such that u ( t 0 ) ∈ N δ 0 + . It follows from Lemma 2.13 that

J ( u 0 ) = l = d δ 1 = d δ 2 < d δ , δ ∈ ( δ 1 , δ 2 ) ,

which combined (4.3) shows that

(5.2) J ( u ( t 0 ) ) ≤ ∫ 0 t 0 ‖ u t ( τ ) ‖ 2 2 d τ + J ( u ( t 0 ) ) = J ( u 0 ) < d δ , δ ∈ ( δ 1 , δ 2 ) .

Moreover, it follows from the fact u 0 ∈ N δ 0 + and the definition of d δ that J ( u ( t 0 ) ) ≥ d δ 0 , which contradicts (5.2).

(ii) Repeating the similar argument as Proof (i), one can show Result (ii) immediately.□

Lemma 5.3

Let u 0 ∈ N − with J ( u 0 ) < d and u be the weak solution of Problems (1.1)–(1.3) corresponding to u 0 , then there exists a constant γ 1 > γ * ≔ S − p + 1 p − 1 such that

(5.3) ‖ ∇ E u ( t ) ‖ 2 ≥ γ 1 a n d ‖ u ( t ) ‖ p + 1 ≥ S γ 1 , f o r a l l t ≥ 0 .

In addition, if J ( u 0 ) ≥ 0 , then

(5.4) γ 1 γ * ≥ ( p + 1 ) 1 2 − J ( u 0 ) γ * 2 1 p − 1 > 1 .

Proof

To prove (5.3), we define the auxiliary function j ( ⋅ ) as follows:

(5.5) j ( γ ) ≔ 1 2 γ 2 − 1 p + 1 S p + 1 γ p + 1 , γ ∈ ( 0 , ∞ ) .

A simple calculation shows that j ( γ ) is increasing for γ ∈ ( 0 , γ * ) , decreasing for γ ∈ ( γ * , ∞ ) , and take the maximum at γ = γ * , which gives j ( γ * ) = d via (i) in Lemma 2.13. Hence, there exists a constant γ 1 > γ * such that j ( γ 1 ) = J ( u 0 ) due to J ( u 0 ) < d and lim γ → ∞ j ( γ ) = − ∞ . Moreover, it follows from ‖ u 0 ‖ p + 1 ≤ S ‖ ∇ E u 0 ‖ 2 that

(5.6) j ( γ 1 ) = J ( u 0 ) = 1 2 ‖ ∇ E u 0 ‖ 2 2 − 1 p + 1 ‖ u 0 ‖ p + 1 p + 1 ≥ 1 2 ‖ ∇ E u 0 ‖ 2 2 − 1 p + 1 ‖ ∇ E u 0 ‖ 2 p + 1 = j ( ‖ ∇ E u 0 ‖ 2 ) .

From u 0 ∈ N − and (ii) in Lemma 2.12, we obtain ‖ ∇ E u 0 ‖ 2 ≥ γ * by taking δ = 1 directly, which together (5.6) with the fact that j ( r ) is decreasing on ( γ * , ∞ ) shows that ‖ ∇ E u 0 ‖ 2 ≥ γ 1 , i.e., ‖ ∇ E u ( t ) ‖ 2 ≥ γ 1 holds for t = 0 . Next, we deduce ‖ ∇ E u ( t ) ‖ 2 ≥ γ 1 holds for all t > 0 . Arguing by contradiction, we assume ‖ ∇ E u ( t 0 ) ‖ 2 < γ 1 for some t 0 > 0 . By the continuity u ∈ C ( [ 0 , T ] ; ℋ 2 , 0 1 , N + 1 2 ( E ) ) , we can choose a t 1 ∈ ( 0 , t 0 ) such that γ * < ‖ ∇ E u ( t 1 ) ‖ 2 < γ 1 , then it follows from the monotonicity properties of j ( γ ) that

J ( u 0 ) = j ( γ 1 ) < j ( ‖ ∇ E u ( t 1 ) ‖ 2 ) ≤ J ( u ( t 1 ) ) ,

which contradicts (4.3), hence, we obtain ‖ ∇ E u ( t ) ‖ 2 ≥ γ 1 for all t > 0 . In addition, it follows from (4.3) and the definition of J ( u ) that

1 p + 1 ‖ u ( t ) ‖ p + 1 p + 1 = − J ( u ( t ) ) + 1 2 ‖ ∇ E u ( t ) ‖ 2 2 ≥ − J ( u 0 ) + 1 2 ‖ ∇ E u ‖ 2 2 ≥ − j ( γ 1 ) + 1 2 γ 1 2 = 1 2 γ 1 2 − 1 2 γ 1 2 + 1 p + 1 S p + 1 γ 1 p + 1 = 1 p + 1 S p + 1 γ 1 p + 1 ,

i.e., (5.3) holds for all t ≥ 0 .

Finally, let β ≔ γ 1 γ * > 1 . Due to J ( u 0 ) ≥ 0 , we infer from (5.5) that,

1 2 − 1 p + 1 β p − 1 = 1 2 − 1 p + 1 S p + 1 ( β γ * ) p − 1 = j ( β γ * ) ( β γ * ) 2 = J ( u 0 ) ( β γ * ) 2 ≤ J ( u 0 ) γ * 2 ,

which yields β = γ 1 γ * ≥ ( p + 1 ) 1 2 − J ( u 0 ) γ * 2 1 p − 1 . Furthermore, one deduces by J ( u 0 ) < d and (i) in Lemma 2.13 that

( p + 1 ) 1 2 − J ( u 0 ) γ * 2 > ( p + 1 ) 1 2 − d γ * 2 = 1 ,

i.e., (5.4) holds.□

Proof of Theorem 3.3

(i) Global existence. Since the conclusion is trivial when u 0 = 0 , we only consider the case u 0 ∈ N + \ { 0 } . It follows from (2.2), (2.3), and (4.3) that the weak solution u given by Theorem 3.1 satisfies

(5.7) J ( u ( t ) ) = p − 1 2 ( p + 1 ) ‖ ∇ E u ( t ) ‖ 2 2 + 1 p + 1 I ( u ( t ) )

and

(5.8) ∫ 0 t ‖ u t ( τ ) ‖ 2 2 d τ + J ( u ( t ) ) = J ( u 0 ) < d , for all t ∈ [ 0 , T max ) .

Then, by Lemma 5.2 and the fact that u 0 ∈ N + and 0 < J ( u 0 ) < d , we deduce that

u ( t ) ∈ N δ + , for all 0 < t < T max and δ 1 < δ < δ 2 .

Moreover, by (iv) in Lemma 2.13, we have 1 ∈ ( δ 1 , δ 2 ) , which implies u ( t ) ∈ N + and

(5.9) I ( u ( t ) ) > 0 , for all t ∈ ( 0 , T max ) .

Thus, the combination of (5.7)–(5.9) shows that

(5.10) ‖ ∇ E u ( t ) ‖ 2 2 + ∫ 0 t ‖ u t ( τ ) ‖ 2 2 d τ ≤ 2 ( p + 1 ) p − 1 J ( u 0 ) < 2 ( p + 1 ) p − 1 d , t ∈ ( 0 , T max ) .

Therefore, the continuation principle yields T max = ∞ , i.e., u 0 ∈ G and the global weak solution u ∈ L ∞ ( 0 , ∞ ; ℋ 2 , 0 1 , N + 1 2 ( E ) ) with u t ∈ L 2 ( 0 , ∞ ; L 2 N + 1 2 ( E ) ) satisfies the estimate (3.2).

Asymptotic behavior. It follows from (5.10) and Proposition 2.5 that

‖ u ‖ p + 1 p + 1 ≤ S p + 1 ‖ ∇ E u ‖ 2 p + 1 ≤ S p + 1 ‖ ∇ E u ‖ 2 p − 1 ‖ ∇ E u ‖ 2 2 ≤ ϱ ‖ ∇ E u ‖ 2 2 ,

where ϱ ≔ S p + 1 2 ( p + 1 ) p − 1 J ( u 0 ) p − 1 2 < 1 by virtue of (i) in Lemma 2.13 and J ( u 0 ) < d . Then, from Proposition 2.6, we obtain

(5.11) I ( u ( t ) ) = ‖ ∇ E u ‖ 2 2 − ‖ u ‖ p + 1 p + 1 ≥ ( 1 − ϱ ) ‖ ∇ E u ‖ 2 2 ≥ ( 1 − ϱ ) λ 1 ‖ u ‖ 2 2 .

Meanwhile, the combination of equations (4.4) and (5.11) shows that

(5.12) d d t ‖ u ( t ) ‖ 2 2 = − 2 I ( u ( t ) ) ≤ − 2 ( 1 − ϱ ) λ 1 ‖ u ‖ 2 2 , for all t ≥ 0 .

Applying the Gronwall lemma to equation (5.12), we deduce Estimate (3.3) immediately. Hence, u 0 ∈ G 0 holds.

(ii) Finite-time blowup. Due to 0 < J ( u 0 ) < d and u 0 ∈ N − , by Lemma 5.2, we have u ( t ) ∈ N δ − for t ∈ [ 0 , T max ) and δ ∈ ( δ 1 , δ 2 ) . Next, we claim that u 0 ∈ ℬ , i.e., T max = T max ( u 0 ) < ∞ . Arguing by contradiction, assuming T max = ∞ . Thus, for any T > 0 , u ∈ C ( [ 0 , T ] ; ℋ 2 , 0 1 , N + 1 2 ( E ) ) is the solution of (1.1)–(1.3) on [ 0 , T ] × E , which implies

(5.13) ∫ 0 T ‖ u ( τ ) ‖ 2 2 d τ ≤ λ 1 − 1 ∫ 0 T ‖ ∇ E u ( τ ) ‖ 2 2 d τ ≤ λ 1 − 1 T max 0 ≤ τ ≤ T ‖ ∇ E u ( τ ) ‖ 2 2 < ∞ , for all T > 0 .

On the other hand, define the auxiliary function

(5.14) ℱ 1 ( t ) ≔ ∫ 0 t ‖ u ( τ ) ‖ 2 2 d τ , t ≥ 0 .

The combination of equation (4.4) and a simple calculation shows that

(5.15) ℱ 1 ′ ( t ) = ‖ u ( t ) ‖ 2 2 and ℱ 1 ″ ( t ) = 2 ( u t , u ) E = − 2 I ( u ( t ) ) , t ≥ 0 .

From the relationship between (2.2) and (2.3), (4.3), and Proposition 2.9, (5.15) becomes

(5.16) ℱ 1 ″ ( t ) = − 2 ( p + 1 ) J ( u ) + ( p − 1 ) ‖ ∇ E u ‖ 2 2 ≥ 2 ( p + 1 ) ∫ 0 t ‖ u t ( τ ) ‖ 2 2 d τ − J ( u 0 ) + ( p − 1 ) λ 1 ℱ 1 ′ ( t ) .

Note that

∫ 0 t ∫ E w q u t u d σ d τ 2 = 1 2 ∫ 0 t d d τ ‖ u ‖ 2 2 d τ 2 = 1 4 ( ‖ u ( t ) ‖ 2 4 − 2 ‖ u 0 ‖ 2 2 ‖ u ( t ) ‖ 2 2 + ‖ u 0 ‖ 2 4 ) = 1 4 ( ( ℱ 1 ′ ( t ) ) 2 − 2 ℱ 1 ′ ( t ) ‖ u 0 ‖ 2 2 + ‖ u 0 ‖ 2 4 ) ,

then the combination of (5.14)–(5.16) and Proposition 2.8 shows that

(5.17) ℱ 1 ( t ) ℱ 1 ″ ( t ) − p + 1 2 ( ℱ 1 ′ ( t ) ) 2 ≥ ℱ 1 ( t ) 2 ( p + 1 ) ∫ 0 t ‖ u t ‖ 2 2 d τ − J ( u 0 ) + ( p − 1 ) λ 1 ℱ 1 ′ ( t ) − p + 1 2 4 ∫ 0 t ∫ E w q u t u d σ d τ 2 + 2 ℱ 1 ′ ( t ) ‖ u 0 ‖ 2 2 − ‖ u 0 ‖ 2 4 = 2 ( p + 1 ) ∫ 0 t ‖ u ‖ 2 2 d τ ∫ 0 t ‖ u t ‖ 2 2 d τ − ∫ 0 t ∫ E w q u t u d σ d τ 2 − 2 ( p + 1 ) J ( u 0 ) ℱ 1 ( t ) + ( p − 1 ) λ 1 ℱ 1 ( t ) ℱ 1 ′ ( t ) − ( p + 1 ) ℱ 1 ′ ( t ) ‖ u 0 ‖ 2 2 + p + 1 2 ‖ u 0 ‖ 2 4 ≥ ( p − 1 ) λ 1 ℱ 1 ( t ) ℱ 1 ′ ( t ) − ( p + 1 ) ℱ 1 ′ ( t ) ‖ u 0 ‖ 2 2 − 2 ( p + 1 ) J ( u 0 ) ℱ 1 ( t ) + p + 1 2 ‖ u 0 ‖ 2 4 .

In view of (ii) in Lemma 2.12, we have ‖ ∇ E u ( t ) ‖ 2 > r ( δ ) for all t ≥ 0 and δ ∈ ( δ 1 , δ 2 ) . Then, by the continuity of I δ ( u ( t ) ) and r ( δ ) with respect to δ , we obtain I δ 2 ( u ( t ) ) ≤ 0 and ‖ ∇ E u ( t ) ‖ 2 ≥ r ( δ 2 ) for t ≥ 0 . Recalling (5.15), we have

ℱ 1 ″ ( t ) = − 2 I ( u ( t ) ) = − 2 ( ( 1 − δ 2 ) ‖ ∇ E u ( t ) ‖ 2 2 + I δ 2 ( u ) ) ≥ 2 ( δ 2 − 1 ) r 2 ( δ 2 ) , ℱ 1 ′ ( t ) ≥ 2 ( δ 2 − 1 ) r 2 ( δ 2 ) t + ℱ 1 ′ ( 0 ) ≥ 2 ( δ 2 − 1 ) r 2 ( δ 2 ) t , ℱ 1 ( t ) ≥ ( δ 2 − 1 ) r 2 ( δ 2 ) t 2 + ℱ 1 ( 0 ) ≥ ( δ 2 − 1 ) r 2 ( δ 2 ) t 2 .

Therefore, for a sufficiently large time T 1 , we have

(5.18) 1 2 ( p − 1 ) λ 1 ℱ 1 ( t ) ≥ ( p + 1 ) ‖ u 0 ‖ 2 2 and 1 2 ( p − 1 ) λ 1 ℱ 1 ′ ( t ) ≥ 2 ( p + 1 ) J ( u 0 ) , for all t ≥ T 1 .

The combination of (5.17) and (5.18) shows that

(5.19) ℱ 1 ( t ) ℱ 1 ″ ( t ) − p + 1 2 ( ℱ 1 ′ ( t ) ) 2 ≥ 0 , for all t ≥ T 1 .

Let ψ ( t ) = ℱ 1 ( t + T 1 ) for all t ≥ 0 . Obviously,

ψ ( 0 ) = ℱ 1 ( T 1 ) = ∫ 0 T 1 ‖ u ( τ ) ‖ 2 2 d τ > 0 and ψ ′ ( 0 ) = ℱ 1 ′ ( T 1 ) = ‖ u ( T 1 ) ‖ 2 2 > 0 .

Moreover, one can infer from (5.19) that

ψ ( t ) ψ ″ ( t ) − 1 + p − 1 2 ( ψ ′ ( t ) ) 2 ≥ 0 , t > 0 .

By Lemma 2.10, we have

∫ 0 t 1 + T 1 ‖ u ( τ ) ‖ 2 2 d τ = lim t → t 1 ∫ 0 t + T 1 ‖ u ( τ ) ‖ 2 2 d τ = lim t → t 1 ψ ( t ) = ∞ ,

for some 0 < t 1 ≤ 2 ∫ 0 T 1 ‖ u ( τ ) ‖ 2 2 d τ ( p − 1 ) ‖ u ( T 1 ) ‖ 2 2 , which contradicts (5.13); hence, we have T max < ∞ , i.e., u 0 ∈ ℬ .

The upper-bound of blowup time. To prove Estimate (3.4), we denote the auxiliary function

(5.20) ℱ 2 ( t ) ≔ d − J ( u ( t ) ) , t ∈ [ 0 , T max ) .

Coupled with J ( u 0 ) < d and (4.2), (5.20) tells us ℱ 2 ( t ) > 0 for all t ∈ [ 0 , T max ) . Recalling (5.15), together with the representation of J ( u ) and ℱ 2 ( t ) , we derive

(5.21) ℱ 1 ″ ( t ) = − 2 I ( u ( t ) ) = − 2 ‖ ∇ E u ‖ 2 2 + 2 ‖ u ‖ p + 1 p + 1 = 2 ( p − 1 ) p + 1 ‖ u ‖ p + 1 p + 1 − 4 J ( u ( t ) ) = 2 ( p − 1 ) p + 1 ‖ u ‖ p + 1 p + 1 − 4 d + 4 ℱ 2 ( t ) , t ∈ [ 0 , T max ) .

By (5.3) and (i) in Lemma 2.13, we obtain

(5.22) 4 d = 2 ( p − 1 ) p + 1 S − 2 ( p + 1 ) p − 1 = 2 ( p − 1 ) p + 1 γ * γ 1 S γ 1 p + 1 ≤ 2 ( p − 1 ) p + 1 γ * γ 1 ‖ u ( t ) ‖ p + 1 p + 1 ,

for all t ∈ [ 0 , T max ) , where γ * = S − p + 1 p − 1 and γ 1 > γ * are given by Lemma 5.3. Since ℱ 2 ( t ) > 0 , then substituting (5.22) into (5.21), we obtain

(5.23) ℱ 1 ″ ( t ) ≥ 2 ( p − 1 ) p + 1 1 − γ * p + 1 γ 1 p + 1 ‖ u ( t ) ‖ p + 1 p + 1 + 4 ℱ 2 ( t ) > C 0 ‖ u ( t ) ‖ p + 1 p + 1 , t ∈ [ 0 , T max ) ,

where C 0 ≔ 2 ( p − 1 ) p + 1 1 − γ * p + 1 γ 1 p + 1 > 0 . By Proposition 2.5, one obtains

(5.24) ( ℱ 1 ′ ( t ) ) p + 1 2 = ( ‖ u ( t ) ‖ 2 2 ) p + 1 2 ≤ ∣ E ∣ p − 1 2 ‖ u ( t ) ‖ p + 1 p + 1 , t ∈ [ 0 , T max ) .

Combining (5.23) and (5.24), we have

(5.25) ℱ 1 ″ ( t ) > C 0 ∣ E ∣ p − 1 2 ( ℱ 1 ′ ( t ) ) p + 1 2 , t ∈ [ 0 , T max ) ,

which implies

(5.26) ℱ 1 ′ ( t ) > ( ℱ 1 ′ ( 0 ) ) 1 − p 2 − C 0 ( p − 1 ) 2 ∣ E ∣ p − 1 2 t − 2 p − 1 = ‖ u 0 ‖ 2 1 − p − C 0 ( p − 1 ) 2 ∣ E ∣ p − 1 2 t − 2 p − 1 ,

for all t ∈ [ 0 , T max ) . Thus, we obtain that ℱ 1 ′ ( t ) = ‖ u ( t ) ‖ 2 2 blows up at some finite-time. Moreover, the combination of equations (5.26) and (5.4) shows that

T max ≤ ( p + 1 ) ∣ E ∣ p − 1 2 ( p − 1 ) 2 1 − γ * p + 1 γ 1 p + 1 ‖ u 0 ‖ 2 1 − p ≤ ( p + 1 ) ∣ E ∣ p − 1 2 ( p − 1 ) 2 1 − ( p + 1 ) 1 2 − J ( u 0 ) γ * 2 − p + 1 p − 1 ‖ u 0 ‖ 2 1 − p ,

i.e., Estimate (3.4) holds.

The upper-bound of blowup rate. Integrating (5.25) from t to T , then by ℱ 1 ′ ( T ) = + ∞ , we derive

‖ u ( t ) ‖ 2 2 = ℱ 1 ′ ( t ) < C 0 ( p − 1 ) 2 ∣ E ∣ p − 1 2 2 1 − p ( T max − t ) − 2 p − 1 = 2 p − 1 2 ( p − 1 ) 2 ( p + 1 ) ∣ E ∣ p − 1 2 1 − γ * γ 1 p + 1 2 1 − p ( T max − t ) 2 1 − p ≤ ( p − 1 ) 4 1 − p ∣ E ∣ 2 ( p + 1 ) 2 1 − p 1 − ( p + 1 ) 1 2 − J ( u 0 ) γ * 2 − p + 1 p − 1 2 1 − p ( T max − t ) 2 1 − p .

Then, we obtain Estimate (3.5).

Asymptotic behavior of blowup solutions. Let

ℱ 3 ( t ) ≔ ℱ 2 ( t ) + 1 2 ℱ 1 ′ ( t ) , for all t ∈ [ 0 , T max ) .

Then, by (4.2) and (5.15), we obtain

(5.27) ℱ 3 ′ ( t ) = − d d t J ( u ( t ) ) + 1 2 ℱ 1 ″ ( t ) = ‖ u t ( t ) ‖ 2 2 − I ( u ( t ) ) , t ∈ [ 0 , T max ) .

Using the definition of J ( u ) , I ( u ) , and ℱ 2 ( t ) , we obtain

(5.28) I ( u ) = ( p + 1 ) J ( u ) − p − 1 2 ‖ ∇ E u ‖ 2 2 = ( p + 1 ) d − ( p + 1 ) ℱ 2 ( t ) − p − 1 2 ‖ ∇ E u ‖ 2 2 .

It follows from (5.3) that

(5.29) ‖ ∇ E u ‖ 2 2 = γ 1 2 − γ * 2 γ 1 2 ‖ ∇ E u ‖ 2 2 + γ * 2 γ 1 2 ‖ ∇ E u ‖ 2 2 ≥ γ 1 2 − γ * 2 γ 1 2 ‖ ∇ E u ‖ 2 2 + γ * 2 .

Then, combining (5.27)–(5.29) and d = p − 1 2 ( p + 1 ) γ * 2 , we have

ℱ 3 ′ ( t ) = ‖ u t ( t ) ‖ 2 2 − ( p + 1 ) d + ( p + 1 ) ℱ 2 ( t ) + p − 1 2 ‖ ∇ E u ( t ) ‖ 2 2 ≥ ( p + 1 ) ℱ 2 ( t ) + p − 1 2 γ 1 2 − γ * 2 γ 1 2 ‖ ∇ E u ( t ) ‖ 2 2 , t ∈ [ 0 , T max ) .

For any t ∈ [ 0 , T max ) , since ℱ 2 ( t ) > 0 , we have

(5.30) ℱ 3 ′ ( t ) ≥ C 1 ( ℱ 2 ( t ) + ‖ ∇ E u ( t ) ‖ 2 2 ) with C 1 = min p + 1 , p − 1 2 γ 1 2 − γ * 2 γ 1 2 > 0 .

On the other hand, using Proposition 2.6, we have

ℱ 3 ( t ) = ℱ 2 ( t ) + 1 2 ‖ u ( t ) ‖ 2 2 ≤ ℱ 2 ( t ) + 1 2 λ 1 ‖ ∇ E u ( t ) ‖ 2 2 , t ∈ [ 0 , T max ) .

Taking C 2 = max { 1 , 1 2 λ 1 2 } > 0 , then

(5.31) ℱ 3 ( t ) ≤ C 2 ( ℱ 2 ( t ) + ‖ ∇ E u ( t ) ‖ 2 2 ) , t ∈ [ 0 , T max ) .

Combining (5.30) and (5.31), we obtain

ℱ 3 ′ ( t ) ≥ C 1 C 2 ℱ 3 ( t ) , for t ∈ [ 0 , T max ) .

Applying the Gronwall inequality to the aforementioned estimate shows that

(5.32) ℱ 3 ( t ) ≥ ℱ 3 ( 0 ) e C 1 C 2 t , for t ∈ [ 0 , T max ) .

By the definition of ℱ 3 ( t ) , J ( u ) , and Proposition 2.8, we have

ℱ 3 ( t ) = d − 1 2 ‖ ∇ E u ‖ 2 2 + 1 p + 1 ‖ u ‖ p + 1 p + 1 + 1 2 ‖ u ‖ 2 2 ≤ d + 1 p + 1 ‖ u ‖ p + 1 p + 1 + 1 2 ∣ E ∣ p − 1 p + 1 ‖ u ‖ p + 1 2 ,

which implies

‖ u ‖ p + 1 p + 1 + ‖ u ‖ p + 1 2 ≥ C 3 − 1 ( ℱ 3 ( t ) − d ) with C 3 = max 1 p + 1 , 1 2 ∣ E ∣ p − 1 p + 1 > 0 .

Hence, coupled with (5.32) and the definition of ℱ 3 ( t ) , it follows that

‖ u ‖ p + 1 p + 1 + ‖ u ‖ p + 1 2 ≥ d − J ( u 0 ) + 1 2 ‖ u 0 ‖ 2 2 C 3 e C 1 C 2 t − d C 3 , for t ∈ [ 0 , T max ) ,

i.e., the solution u ( t ) grows as an exponential function in L p + 1 N + 1 p + 1 ( E ) -norm, which completes the proof.□

Next, we prove Theorem 3.4, which describes the vacuum isolation of the solution to Problems (1.1)–(1.3).

Proof of Theorem 3.4

Let u be the solution of Problems (1.1)–(1.3) corresponding to initial datum u 0 . We only need to prove that if u 0 ≠ 0 and J ( u 0 ) ≤ e < d , then for all δ ∈ ( δ 1 , δ 2 ) and t ∈ ( 0 , T max ) , u ( t ) ∉ N δ , i.e., I δ ( u ( t ) ) ≠ 0 . First, we assert that I δ ( u 0 ) ≠ 0 for all δ ∈ ( δ 1 , δ 2 ) . Indeed, if I δ ( u 0 ) = 0 , then it follows from (iv) in Lemma 2.13 that e ≥ J ( u 0 ) ≥ d δ > d δ 1 = d δ 2 , which contradicts the fact that d δ 1 = d δ 2 = e . Next, we claim that u ( t ) ∈ U e for all t ∈ ( 0 , T max ) . Arguing by contradiction, we assume that there exists a t 1 : t 1 ∈ ( 0 , T max ) and a δ ∈ ( δ 1 , δ 2 ) such that u ( t 1 ) ∈ N δ . It follows from (4.3) that

J ( u 0 ) ≥ J ( u ( t 1 ) ) ≥ d δ > d δ 1 = d δ 2 = e ≥ J ( u 0 ) ,

which leads to a contradiction.□

Remark 5.4

By Lemma 2.13, we know that δ 1 decreases to 0 and δ 2 increases to p + 1 2 as e decrease to 0, which implies the vacuum region U e expands as e decreases. As the limit case, we infer the vacuum region for the nontrivial solutions with J ( u 0 ) = 0 is

U 0 = lim e → 0 U e = u ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) ∣ 0 < ‖ ∇ E u ( t ) ‖ 2 < S − p + 1 p − 1 p + 1 2 1 p − 1 .

6 Nonpositive initial energy

Lemma 6.1

Let u be a solution of Problems (1.1)–(1.3) with J ( u 0 ) ≤ 0 , then we have u ( t ) ∈ N − and ‖ ∇ E u ( t ) ‖ 2 > 0 for all t ∈ [ 0 , T max ) .

Proof

Since J ( u 0 ) ≤ 0 , it follows from (4.2) that J ( u ( t ) ) ≤ J ( u 0 ) ≤ 0 for any t ∈ [ 0 , T max ) . Combining the definition of J ( u ) and I ( u ) , we have

(6.1) J ( u ( t ) ) = p − 1 2 ( p + 1 ) ‖ ∇ E u ( t ) ‖ 2 2 + 1 p + 1 I ( u ( t ) ) ≤ 0 ,

which gives u ( t ) ∈ N − and ‖ ∇ E u ( t ) ‖ 2 > 0 for all t ∈ [ 0 , T max ) , the remainder derived by (ii) and (iv) in Lemma 2.12 immediately.□

Proof of Theorem 3.2

(i) According to (4.2) and J ( u 0 ) ≤ 0 , we know J ( u ( t ) ) ≤ J ( u 0 ) ≤ 0 for all t ∈ [ 0 , T max ) . It follows from Formula (2.2) that

1 p + 1 S p + 1 ‖ ∇ E u ( t ) ‖ 2 p + 1 ≥ 1 p + 1 ‖ u ( t ) ‖ p + 1 p + 1 ≥ 1 2 ‖ ∇ E u ( t ) ‖ 2 2 − J ( u 0 ) ,

which implies ℱ ( ‖ ∇ E u ( t ) ‖ 2 ) ≥ 0 for all t ∈ [ 0 , T max ) , where ℱ is defined by:

ℱ ( r ) = 1 p + 1 S p + 1 r p + 1 − 1 2 r 2 + J ( u 0 ) , r ≥ 0 .

Obviously, the hypothesis p > 1 implies that the equation ℱ ( r ) = 0 admits a unique positive root r * = r * ( J ( u 0 ) ) such that ℱ ( r ) ≥ 0 if and only if r ≥ r * ( J ( u 0 ) ) . Thus, we know that ‖ ∇ E u ( t ) ‖ 2 ≥ r * for every t ∈ [ 0 , T max ) . Therefore, the set U r * denoted by (3.1) is a vacuum region for the solution u , i.e., u ( t ) ∉ U r * for all t ∈ [ 0 , T max ) .

(ii) If J ( u 0 ) < 0 , we define

(6.2) ℱ 4 ( t ) ≔ − 2 ( p + 1 ) J ( u ( t ) ) , t ∈ [ 0 , T max ) .

According to (4.2) and p > 1 , we obtain

(6.3) ℱ 4 ′ ( t ) = − 2 ( p + 1 ) d d t J ( u ( t ) ) = 2 ( p + 1 ) ‖ u t ‖ 2 2 ≥ 0 , t ∈ [ 0 , T max ) ,

which implies that ℱ 4 ( t ) ≥ ℱ 4 ( 0 ) = − 2 ( p + 1 ) J ( u 0 ) > 0 for any t ∈ [ 0 , T max ) . It follows form (5.15) and Lemma 6.1 that

(6.4) ℱ 1 ″ ( t ) = − 2 I ( u ( t ) ) = ( p − 1 ) ‖ ∇ E u ( t ) ‖ 2 2 + ℱ 4 ( t ) > ℱ 4 ( t ) > 0 , t ∈ [ 0 , T max ) ,

which implies ℱ 1 ′ ( t ) > 0 for all t ∈ ( 0 , T max ) . Combining (6.3), Schwartz’s inequality, and (6.4), for any t ∈ ( 0 , T max ) , we obtain

1 2 ( p + 1 ) ℱ 1 ′ ( t ) ℱ 4 ′ ( t ) = ‖ u ( t ) ‖ 2 2 ‖ u t ( t ) ‖ 2 2 ≥ ( u ( t ) , u t ( t ) ) E 2 = 1 4 ( ℱ 1 ″ ( t ) ) 2 > 1 4 ℱ 1 ″ ( t ) ℱ 4 ( t ) ,

which implies

(6.5) ℱ 4 ′ ( t ) ℱ 4 ( t ) > p + 1 2 ℱ 1 ″ ( t ) ℱ 1 ′ ( t ) , t ∈ ( 0 , T max ) .

Integrating (6.5) from 0 to t and applying (6.4), we have

(6.6) ℱ 1 ″ ( t ) ( ℱ 1 ′ ( t ) ) p + 1 2 > ℱ 4 ( t ) ( ℱ 1 ′ ( t ) ) p + 1 2 > ℱ 4 ( 0 ) ( ℱ 1 ′ ( 0 ) ) p + 1 2 , t ∈ ( 0 , T max ) .

Integrating (6.6) from 0 to t , we see

(6.7) 1 ( ℱ 1 ′ ( t ) ) p − 1 2 < 1 ( ℱ 1 ′ ( 0 ) ) p − 1 2 − p − 1 2 ℱ 4 ( 0 ) ( ℱ 1 ′ ( 0 ) ) p + 1 2 t , t ∈ ( 0 , T max ) .

Obviously, it follows from (6.7) and (5.15) that

0 ≤ ‖ u ( t ) ‖ 2 − ( p − 1 ) ≤ 1 ( ℱ 1 ′ ( 0 ) ) p − 1 2 − p − 1 2 ℱ 4 ( 0 ) ( ℱ 1 ′ ( 0 ) ) p + 1 2 t → 0 as t → 2 p − 1 ℱ 1 ′ ( 0 ) ℱ 4 ( 0 ) ,

which implies

lim t → T max ‖ u ( t ) ‖ 2 = ∞ and T max ≤ 2 p − 1 ℱ 1 ′ ( 0 ) ℱ 4 ( 0 ) = ‖ u 0 ‖ 2 2 ( 1 − p 2 ) J ( u 0 ) .

Due to the fact that ℱ 1 ′ ( t ) blows up in finite-time T max , integrating (6.6) from t to T max , we arrive at

ℱ 1 ′ ( t ) < ( T max − t ) 2 1 − p ( p − 1 ) ℱ 4 ( 0 ) 2 ( ℱ 1 ′ ( 0 ) ) p + 1 2 2 1 − p ,

then it follows from the definition of ℱ 1 ′ ( t ) and ℱ 4 ( t ) that

‖ u ( t ) ‖ 2 ≤ ( 1 − p 2 ) J ( u 0 ) ‖ u 0 ‖ 2 p + 1 1 1 − p ( T max − t ) 1 1 − p .

If J ( u 0 ) = 0 , similar to the proof of (ii) in Theorem 3.3, we obtain the following inequality by (5.17) that

(6.8) ℱ 1 ( t ) ℱ 1 ″ ( t ) − p + 1 2 ( ℱ 1 ′ ( t ) ) 2 ≥ ( p − 1 ) λ 1 ℱ 1 ( t ) ℱ 1 ′ ( t ) − ( p + 1 ) ℱ 1 ′ ( t ) ‖ u 0 ‖ 2 2 .

By Lemma 6.1, we obtain I ( u ( t ) ) < 0 for t ∈ [ 0 , T max ) . Thus, we know ℱ 1 ″ ( t ) = − 2 I ( u ( t ) ) > 0 for t ∈ [ 0 , T max ) in (5.15), which means that ℱ 1 ′ ( t ) is increasing with respect to t . Since ℱ 1 ′ ( 0 ) = ‖ u 0 ‖ 2 2 ≥ 0 , we know that there exists t 0 > 0 such that ℱ 1 ′ ( t 0 ) > 0 and

ℱ 1 ( t ) ≥ ℱ 1 ′ ( t 0 ) ( t − t 0 ) + ℱ 1 ( t 0 ) ≥ ℱ 1 ′ ( t 0 ) ( t − t 0 ) , t ∈ [ t 0 , T max ) ,

which implies

( p − 1 ) λ 1 ℱ 1 ( t ) > ( p + 1 ) ‖ u 0 ‖ 2 2 , t > t 1 ≔ λ 1 ( p + 1 ) p − 1 ‖ u 0 ‖ 2 2 ‖ u ( t 0 ) ‖ 2 2 + t 0 .

Then, from (6.8), we have

ℱ 1 ( t ) ℱ 1 ″ ( t ) − p + 1 2 ( ℱ 1 ′ ( t ) ) 2 > 0 , t ∈ [ t 1 , T max ) .

The remainder proof is similar to the proof of (ii) in Theorem 3.3, including the proof of the estimate of the blowup time and blowup rate; hence, we omit it and conclude that the solution blows up in finite-time. In particular, the estimate results of the blowup time and blowup rate can be obtained immediately by taking J ( u 0 ) = 0 in (ii) of Theorem 3.3.□

7 Critical initial energy

First, we prove the global existence of the weak solution.

Proof of Theorem 3.5

(i) Global existence. We know ‖ ∇ E u 0 ‖ 2 ≠ 0 via J ( u 0 ) = d . Choice a sequence { λ m } that satisfies 0 < λ < 1 , m = 1 , 2 , … , and λ m → 1 as m → ∞ . Then, we set u 0 m = λ m u 0 and investigate the following new initial boundary value problem:

(7.1) u t − Δ E u = ∣ u ∣ p − 1 u , x ∈ int E , t > 0 ,

(7.2) u ( x , 0 ) = u 0 m ( x ) , x ∈ int E ,

(7.3) u ( x , t ) = 0 , x ∈ ∂ E , t ≥ 0 .

From I ( u 0 ) ≥ 0 and Lemma 2.11, we have λ * = λ * ( u 0 ) ≥ 1 . Then, we obtain I ( u 0 m ) = I ( λ m u 0 ) > 0 and J ( u 0 m ) = J ( λ m u 0 ) < J ( u 0 ) = d . Thus, by Theorem 3.3, it follows that for every m Problems (7.1)–(7.3) admit a global solution u m ∈ L ∞ ( 0 , ∞ ; ℋ 2 , 0 1 , N + 1 2 ( E ) ) with u m t ∈ L 2 ( 0 , ∞ ; L 2 N + 1 2 ( E ) ) and u m ∈ N + for all t ≥ 0 . Indeed, we can repeat the proof in Theorem 3.3 as long as we replace the u m 0 by u m 0 n and take the limit u m n → u to show the existence of a global solution, which means u 0 ∈ G provided u 0 ∈ N + and J ( u 0 ) = d .

Asymptotic behavior of solution. Let u be a global solution of Problems (1.1)–(1.3) with J ( u 0 ) = d and I ( u 0 ) > 0 , then we assert u ( t ) ∈ N + for t ≥ 0 . Arguing by contradiction, if there exists a first time t * > 0 such that I ( u ( t * ) ) = 0 and I ( u ( t ) ) > 0 for t ∈ [ 0 , t * ) , then from the definition of d , we deduce J ( u ( t * ) ) ≥ d . By (4.3), it follows

d ≤ J ( u ( t * ) ) = d − ∫ 0 t * ‖ u t ( τ ) ‖ 2 d τ ≤ d ,

for any t * > 0 . Thus, we obtain J ( u ( t * ) ) = d , which means

∫ 0 t * ‖ u t ( τ ) ‖ 2 d τ = 0 ,

i.e., ‖ u t ( t ) ‖ 2 ≡ 0 for 0 ≤ t ≤ t * , which contradicts I ( u 0 ) > 0 . Hence, we conclude u ( t ) ∈ N + for all t ≥ 0 .

According to the proof in Theorem 3.3, we need to make minor changes to the initial energy to guarantee the repeatability of the proof process. By the continuity of J ( u ) and I ( u ) with respect to t , we can choose any t 1 ≥ 0 as the initial time, then u ( t ) ∈ N + for all t ≥ t 1 , then we have J ( u ( t 1 ) ) < d . Thus, we claim that the solution decays as exponential by Theorem 3.3, i.e., u 0 ∈ G 0 .

Decay estimate of energy. Let u be a global solution of Problems (1.1)–(1.3) with J ( u 0 ) = d and I ( u 0 ) > 0 , then by the aforementioned proof, we also infer u ( t ) ∈ N + for t ≥ 0 . Together with (4.2) and the definition of d , we arrive at

d > J ( u ( t 1 ) ) ≥ J ( u ( t ) ) ≥ p − 1 2 ( p + 1 ) ‖ ∇ E u ( t ) ‖ 2 2 , t ≥ t 1 > 0 ,

and furthermore,

(7.4) ‖ u ‖ p + 1 p + 1 ≤ S p + 1 ‖ ∇ E u ‖ 2 p − 1 ‖ ∇ E u ‖ 2 2 ≤ S p + 1 2 ( p + 1 ) p − 1 J ( u ( t 1 ) ) p − 1 2 ‖ ∇ E u ‖ 2 2 ≤ J ( u ( t 1 ) ) d p − 1 2 ‖ ∇ E u ‖ 2 2 , t ≥ t 1 > 0 .

Setting ℱ 5 ( t ) ≔ ‖ u ( t ) ‖ p + 1 p + 1 ‖ ∇ E u ( t ) ‖ 2 2 for t ≥ t 1 , then we deduce from (7.4) that ℱ 5 ( t ) ∈ [ 0 , ρ ] ⊂ [ 0 , 1 ) for all t ≥ t 1 , where ρ ≔ J ( u ( t 1 ) ) d p − 1 2 . Again from I ( u ( t ) ) > 0 for t 1 ≤ t < ∞ , we derive

J ( u ( t ) ) I ( u ( t ) ) = 1 2 ‖ ∇ E u ( t ) ‖ 2 2 − 1 p + 1 ‖ u ( t ) ‖ p + 1 p + 1 ‖ ∇ E u ( t ) ‖ 2 2 − ‖ u ( t ) ‖ p + 1 p + 1 = p + 1 − 2 ℱ 5 ( t ) 2 ( p + 1 ) ( 1 − ℱ 5 ( t ) ) , t ≥ t 1 .

Note that the function f ( ξ ) ≔ p + 1 − 2 ξ 2 ( p + 1 ) ( 1 − ξ ) is increasing with respect to ξ on [ 0 , ρ ] , and we easily deduce that f ( ξ ) ∈ 1 2 , p + 1 − 2 ρ 2 ( p + 1 ) ( 1 − ρ ) for all ξ ∈ [ 0 , ρ ] ; thus,

(7.5) J ( u ( t ) ) I ( u ( t ) ) ∈ 1 2 , p + 1 − 2 ρ 2 ( p + 1 ) ( 1 − ρ ) , t ≥ t 1 .

In addition, (4.2) implies that

(7.6) d d t ( t J ( u ( t ) ) ) ≤ J ( u ( t ) ) , t ≥ 0 .

Integrating (7.6) from t 1 to t , we conclude from (7.5) and (4.4) that

t J ( u ( t ) ) − t 1 J ( u ( t 1 ) ) ≤ ∫ t 1 t J ( u ( τ ) ) d τ ≤ p + 1 − 2 ρ 2 ( p + 1 ) ( 1 − ρ ) ∫ t 1 t I ( u ( τ ) ) d τ ≤ − p + 1 − 2 ρ 4 ( p + 1 ) ( 1 − ρ ) ∫ t 1 t d d t ‖ u ( τ ) ‖ 2 2 d τ ≤ p + 1 − 2 ρ 4 ( p + 1 ) ( 1 − ρ ) ‖ u ( t 1 ) ‖ 2 2 ,

for t ≥ t 1 , which yields

J ( u ( t ) ) ≤ p + 1 − 2 ρ 4 ( p + 1 ) ( 1 − ρ ) ‖ u ( t 1 ) ‖ 2 2 + t 1 J ( u ( t 1 ) ) t − 1 , t ≥ t 1 .

(ii) Finite-time blowup. Let u be a solution of Problems (1.1)–(1.3) with J ( u 0 ) = d and I ( u 0 ) < 0 . We are going to prove T max < ∞ . According to the continuity of J ( u ) and I ( u ) with respect to t , we infer that there exists a sufficient small t 1 > 0 such that J ( u ( t 1 ) ) > 0 and I ( u ( t ) ) < 0 for t ∈ [ 0 , t 1 ] . Therefore, we deduce by (5.15) that ℱ 1 ″ ( t ) = − 2 I ( u ) > 0 and ‖ u t ( t ) ‖ 2 ≢ 0 for t ∈ [ 0 , t 1 ] . Hence, combining (4.3), we pick up such a t 1 that

0 < J ( u ( t 1 ) ) = d − ∫ 0 t 1 ‖ u t ( τ ) ‖ 2 2 d τ = d 1 < d .

Choosing t = t 1 as the initial time and by (ii) in Lemma 5.2, we obtain u ( t ) ∈ N − for t > t 1 . The remainder proof is similar as the proof in (ii) in Theorem 3.3, which eventually implies that the existence time of the solution u is finite, i.e., lim t → T max − ∫ t 1 t ‖ u ( τ ) ‖ 2 2 d τ = + ∞ .

The upper-bound of blowup time and blowup rate and the asymptotic of blowup solution. Similar to the aforementioned proof in (ii) of Theorem 3.5, we still obtain

J ( u ( t 1 ) ) = d − ∫ 0 t 1 ‖ u t ( τ ) ‖ 2 2 d τ < d .

Taking t = t 1 as the initial time, we have J ( u ( t 1 ) ) < d and I ( u ( t 1 ) ) < 0 . Thus, similar to the proof shown in (ii) of Theorem 3.3, we remain all results; hence, we omit it.□

8 Supercritical initial energy

First of all, we focus on the stationary problem for Problems (1.1)–(1.3)

(8.1) − Δ E u = ∣ u ∣ p − 1 u , ( ω , x , y ) ∈ E , u = 0 , ( ω , x , y ) ∈ ∂ E .

For any u 0 ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) , there exists a maximal solution of Problems (1.1)–(1.3) defined on [ 0 , T ) denoted by S ( t ) u 0 ≡ u ( t , u 0 ) . Suppose that sup t > 0 ‖ S ( t ) u 0 ‖ ℋ 1 ≤ M for some M > 0 . Then, { S ( t ) u 0 } t > 0 is a relatively compact set in ℋ 2 , 0 1 , N + 1 2 ( E ) , which is a consequence of the fact that the operator e i θ Δ E has compact resolvent [7,8]. Therefore, instead of u ( t ) , we write the solution of Problems (1.1)–(1.3) as u ( t ) = S ( t ) u 0 in this section. Furthermore, if T max ( u 0 ) = ∞ , we denote by

ω ( u 0 ) ≔ ⋂ t ≥ 0 ⋃ s ≥ t S ( t ) u 0 ¯

the ω -limit set of u 0 ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) .

Lemma 8.1 is occupied with the proof of the classification of initial data for global existence and nonexistence of solutions at high initial energy. A rather close proof of this can be found in [24]. Obviously, the edge-degenerate operator appearing in our stationary Problem (8.1) makes the discussion must lie in the weighted Sobolev space ℋ 2 , 0 1 , N + 1 2 ( E ) , so we need to guarantee that the new space also has the technical tools needed for the argument in [24]. Fortunately, a comprehensive result in Corollary 4.2 has already shown that the comparison principle holds, which implies that we can obtain the following lemmas by suitably combining the proof of Gazzola and Weth [24] via a slight modification reinterpreting them in the light of our results about the nonlinear semigroup and comparison principle.

Lemma 8.1

Let v ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) be a nontrivial solution of (8.1), and u 0 ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) , u 0 ≢ ± v .

  1. If v + ≔ max { v ( w , x , y ) , 0 } ≢ 0 and u 0 ≥ v , then u 0 ∈ ℬ ;

  2. If v − ≔ min { v ( w , x , y ) , 0 } ≢ 0 and u 0 ≤ v , then u 0 ∈ ℬ ;

  3. If 0 ≤ u 0 ≤ v , then u 0 ∈ G 0 .

Proof of Theorem 3.6

(i) Take M > 0 and v is a positive solution for Problems (8.1), and let

E ′ = { ( w , x , y ) ∈ E ∣ v ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) , v > ε } ⊂ E ,

be an open subset for a sufficiently small ε > 0 . We can choose a function ϕ h ∈ C 0 1 ( E ′ ) satisfying ϕ h > 0 for all h > 0 and induce a continuous zero extension to E \ E ′ , which satisfies

‖ ∇ E ϕ h ‖ 2 ≥ h and ‖ ϕ h ‖ ∞ ≤ ε ,

where ‖ ⋅ ‖ ∞ denotes the norm of function, i.e., essentially bounded on E . For a fixed h > 0 , it is easy to put ϱ + ≔ v + ϕ h and ϱ − ≔ v − ϕ h . Hence, we obtain that ϱ ± ≥ 0 a.e. in E gives

∫ E ′ w q ∣ ϱ ± ∣ p k + 1 d σ ≤ ∫ E ′ w q ∣ v ∣ p k + 1 d σ + ∫ E ′ w q ∣ ϕ h ∣ p k + 1 d σ ≤ ∫ E ′ w q ∣ v ∣ p k + 1 d σ + ε p k + 1 ∣ E ′ ∣

and

J ( ϱ ± ) = 1 2 ‖ ∇ E ϱ ± ‖ 2 2 − 1 p + 1 ‖ ϱ ± ‖ p + 1 p + 1 = 1 2 ∫ E ′ w q ∣ ∇ E ϱ ± ∣ 2 d σ + ∫ E \ E ′ ∣ ∇ E ϱ ± ∣ 2 d σ − 1 p + 1 ∫ E ′ w q ∣ ϱ ± ∣ p + 1 d σ + ∫ E \ E ′ ∣ ϱ ± ∣ p + 1 d σ ≥ 1 2 ∫ E ′ w q ∣ ∇ E ϱ ± ∣ 2 d σ − 1 p + 1 ∫ E ′ w q ∣ ϱ ± ∣ p + 1 d σ + ∫ E E ′ ∣ ϱ ± ∣ p + 1 d σ .

Clearly, because v is a positive solution for Problem (8.1) and the continuous extension property of ϕ h ∈ C 0 1 ( E ′ ) , it follows that ∫ E \ E ′ w q ∣ ϱ ± ∣ p + 1 d σ = ∫ E \ E ′ w q ∣ v ∣ p + 1 d σ is bounded in ℋ 2 , 0 1 , N + 1 2 ( E ) and independent of t . Thus,

J ( ϱ ± ) ≥ 1 2 ∫ E ′ w q ∣ ∇ E ϱ ± ∣ 2 d σ − 1 p + 1 ∫ E ′ w q ∣ ϱ ± ∣ p + 1 d σ − C ( ε ) p + 1 ≥ 1 2 h − ∫ E ′ w q ∣ ∇ E v ∣ 2 d σ 1 2 2 − ∫ E ′ w q ∣ v ∣ p k + 1 d σ + ε p k + 1 ∣ E ′ ∣ − C ( ε ) p + 1 ,

where C ( ε ) = ∫ E \ E ′ w q ∣ ϱ ± ∣ p + 1 d σ . Similarly, it also follows that

I ( ϱ ± ) = ‖ ∇ E ϱ ± ‖ 2 2 − ‖ ϱ ± ‖ p + 1 p + 1 = ∫ E ′ w q ∣ ∇ E ϱ ± ∣ 2 d σ + ∫ E \ E ′ ∣ ∇ E ϱ ± ∣ 2 d σ − ∫ E ′ w q ∣ ϱ ± ∣ p + 1 d σ + ∫ E \ E ′ ∣ ϱ ± ∣ p + 1 d σ ≥ ∫ E ′ w q ∣ ∇ E ϱ ± ∣ 2 d σ − ∫ E ′ w q ∣ ϱ ± ∣ p + 1 d σ − C ( ε ) ≥ h − ∫ E ′ w q ∣ ∇ E v ∣ 2 d σ 1 2 2 − ∫ E ′ w q ∣ v ∣ p k + 1 d σ + ε p k + 1 ∣ E ′ ∣ − C ( ε ) .

Thus, ϱ ± ∈ N + naturally holds as long as both J ( ϱ ± ) ≥ M and I ( ϱ ± ) > 0 are satisfied for sufficiently large h . For such a value h , assign u 0 ≔ ϱ − and v 0 ≔ ϱ + . Since 0 ≤ u 0 ≤ v , we have u 0 ∈ G 0 by (iii) in Lemma 8.1. On the other hand, we can derive v ϱ ∈ ℬ by 0 ≤ v ≤ v ϱ and (i) in Lemma 8.1.

(ii) Set M > 0 , and one can make two disjoint open sets E i ( i = 1 , 2 ) as the subdomains of E that are arbitrary. Furthermore, picking v ∈ ℋ 2 , 0 1 , N + 1 2 ( E 1 ) ⊂ ℋ 2 , 0 1 , N + 1 2 ( E ) has any non-zero function. Hence, we can easily verify that ‖ κ v ‖ 2 2 ≥ 2 ( p + 1 ) λ 1 ( p − 1 ) M and J ( κ v ) ≤ 0 for large enough κ > 0 . Freezing the real value κ > 0 and selecting a function v ˜ ∈ ℋ 2 , 0 1 , N + 1 2 ( E 2 ) , we obtain J ( v ˜ ) = M − J ( κ v ) . Therefore, u M ≔ κ v + v ˜ yields

J ( u M ) = 1 2 ∫ E 1 w q ∣ ∇ E κ v ∣ 2 d σ − 1 p + 1 ∫ E 1 w q ∣ κ v ∣ p + 1 d σ + ∫ E 2 w q ∣ ∇ E v ˜ ∣ 2 d σ − 1 p + 1 ∫ E 2 w q ∣ v ˜ ∣ p + 1 d σ = J ( κ v ) ∣ κ v ∈ ℋ 2 , 0 1 , N + 1 2 ( E 1 ) + J ( v ˜ ) ∣ v ˜ ∈ ℋ 2 , 0 1 , N + 1 2 ( E 2 ) = M

and

(8.2) ‖ ∇ E u M ‖ 2 2 ≥ λ 1 ‖ u M ‖ 2 2 = λ 1 ‖ κ v ‖ L 2 N + 1 2 ( E 1 ) 2 + ‖ v ˜ ‖ L 2 N + 1 2 ( E 2 ) 2 ≥ λ 1 ‖ κ v ‖ L 2 N + 1 2 ( E 1 ) 2 ≥ 2 ( p + 1 ) p − 1 J ( u M ) .

On the other hand, by the definition of I ( u ) , we have

2 ( p + 1 ) p − 1 J ( u M ) = 2 ( p + 1 ) p − 1 1 2 ‖ ∇ E u M ‖ 2 2 − 1 p + 1 ‖ u M ‖ p + 1 p + 1 ≥ 2 ( p + 1 ) p − 1 1 2 ‖ ∇ E u M ‖ 2 2 − 1 p + 1 ( ‖ ∇ E u M ‖ 2 2 − I ( u M ) ) ≥ ‖ ∇ E u M ‖ 2 2 + 2 p − 1 I ( u M ) .

Combining with (8.2) it is sufficient to ensure I ( u M ) < 0 . Therefore, u M ∈ N − ∩ ℬ in virtue of Theorem 3.6.

A sufficient condition for initial datum that leads to blowup. According to the relationship between J ( u ) and I ( u ) , it follows that

(8.3) J ( u 0 ) = p − 1 2 ( p + 1 ) ‖ ∇ E u 0 ‖ 2 2 + 1 p + 1 I ( u 0 ) .

Obviously, (3.6) infers that u 0 ∈ N − . Next, we discuss it as the following two cases:

  1. If J ( u 0 ) ≤ d , then we conclude that u 0 ∈ ℬ by Theorem 3.3 (ii) and Theorem 3.5 (ii) immediately.

  2. If J ( u 0 ) > d , for any u ∈ N J ( u 0 ) , we deduce by the definition of N J ( u 0 ) that ‖ ∇ E u ‖ 2 2 ≤ 2 J ( u 0 ) ( p + 1 ) p − 1 . By virtue of the edge-type Poincaré inequality, we obtain

    (8.4) ‖ u ‖ 2 2 ≤ λ 1 − 1 ‖ ∇ E u ‖ 2 2 ≤ 2 J ( u 0 ) ( p + 1 ) λ 1 ( p − 1 ) .

    Choosing Λ J ( u 0 ) = sup { ‖ u ‖ 2 2 ∣ u ∈ N J ( u 0 ) } , then (8.4) implies that

    (8.5) Λ J ( u 0 ) ≤ 2 J ( u 0 ) ( p + 1 ) λ 1 ( p − 1 ) .

    Substituting (3.6) into (8.5), it follows that ‖ u 0 ‖ 2 2 > Λ J ( u 0 ) , which together with Theorem 3.7-(ii) shows that u 0 ∈ ℬ .

The upper-bound of blowup time. Now, we shall show the following estimate of upper-bound of blowup time. Suppose that u be the solution to Problems (1.1)–(1.3). For t ∈ [ 0 , T max ) , we denote

(8.6) ℱ 6 ( t ) ≔ ∫ 0 t ‖ u ( τ ) ‖ 2 2 d τ + ( T − t ) ‖ u 0 ‖ 2 2 + μ ( t + ν ) 2 ,

where μ and ν are two undetermined positive constants. Then, it follows from (4.4), edge-type Poincaré inequality, and (4.3) that

(8.7) ℱ 6 ′ ( t ) = ‖ u ( t ) ‖ 2 2 − ‖ u 0 ‖ 2 2 + 2 μ ( t + ν ) , t ∈ [ 0 , T max )

and

(8.8) ℱ 6 ″ ( t ) = − 2 I ( u ( t ) ) + 2 μ ≥ ( p − 1 ) ‖ ∇ E u ( t ) ‖ 2 2 − 2 ( p + 1 ) J ( u ( t ) ) ≥ λ 1 ( p − 1 ) ‖ u ( t ) ‖ 2 2 − 2 ( p + 1 ) J ( u ( t ) ) ≥ λ 1 ( p − 1 ) ‖ u ( t ) ‖ 2 2 − 2 ( p + 1 ) J ( u 0 ) + 2 ( p + 1 ) ∫ 0 t ‖ u t ( τ ) ‖ 2 2 d τ ≥ 2 ( p + 1 ) λ 1 ( p − 1 ) 2 ( p + 1 ) ‖ u 0 ‖ 2 2 − J ( u 0 ) + ∫ 0 t ‖ u t ( τ ) ‖ 2 2 d τ , t ∈ [ 0 , T max ) .

Furthermore, we obtain ℱ 6 ′ ( t ) ≥ 2 μ ( t + ν ) > 0 via (4.4), which implies ℱ 6 ( t ) ≥ ψ ( 0 ) = T ‖ u 0 ‖ 2 2 + μ ν 2 > 0 for any t ∈ [ 0 , T max ) .

On the other hand, it follows

− 1 4 ( ℱ 6 ′ ( t ) ) 2 = − 1 2 ( ‖ u ( t ) ‖ 2 2 − ‖ u 0 ‖ 2 2 ) + μ ( t + ν ) 2 = ∫ 0 t ‖ u ( τ ) ‖ 2 2 d τ + μ ( t + ν ) 2 ∫ 0 t ‖ u t ( τ ) ‖ 2 2 d τ + μ ︸ I 1 − 1 2 ( ‖ u ( t ) ‖ 2 2 − ‖ u 0 ‖ 2 2 ) + μ ( t + ν ) 2 ︸ I 2 − ( ℱ 6 ( t ) − ( T − t ) ‖ u 0 ‖ 2 2 ) ∫ 0 t ‖ u t ( τ ) ‖ 2 2 d τ + μ .

To estimate the aforementioned inequality clearly, we shall show that I 1 − I 2 > 0 ,

I 1 − I 2 = I 1 − 1 2 ( ‖ u ( t ) ‖ 2 2 − ‖ u 0 ‖ 2 2 ) + μ ( t + ν ) 2 = I 1 − 1 2 ∫ 0 t d d τ ‖ u ( τ ) ‖ 2 2 d τ + μ ( t + ν ) 2 = I 1 − ∫ 0 t ( u , u t ) E d τ + μ ( t + ν ) 2 ≥ I 1 − ∫ 0 t ‖ u ( τ ) ‖ 2 ‖ u t ( τ ) ‖ 2 d τ + μ ( t + ν ) 2 ≥ I 1 − ∫ 0 t ‖ u ( τ ) ‖ 2 d τ ∫ 0 t ‖ u t ( τ ) ‖ 2 d τ + μ ( t + ν ) 2 = I 1 − ( ℱ 1 ( t ) ℱ 7 ( t ) + μ ( t + ν ) ) 2 = ( ( ℱ 1 ( t ) ) 2 + μ ( t + ν ) 2 ) ( ( ℱ 7 ( t ) ) 2 + μ ) − ( ℱ 1 ( t ) ℱ 7 ( t ) + μ ( t + ν ) ) 2 = ( μ ℱ 1 ( t ) ) 2 − 2 μ ℱ 1 ( t ) μ ( t + ν ) ℱ 7 ( t ) + ( μ ( t + ν ) ℱ 7 ( t ) ) 2 = ( μ ℱ 1 ( t ) − μ ( t + ν ) ℱ 7 ( t ) ) 2 ≥ 0 ,

where ℱ 7 ( t ) ≔ ∫ 0 t ‖ u t ( τ ) ‖ 2 d τ . Hence, we have

(8.9) − ( ℱ 6 ′ ( t ) ) 2 ≥ − 4 ( ℱ 6 ( t ) − ( T − t ) ‖ u 0 ‖ 2 2 ) ( ℱ 7 ( t ) + μ ) ≥ − 4 ℱ 6 ( t ) ( ℱ 7 ( t ) + μ ) .

Then by (8.6), (8.8), and (8.9), we achieve

ℱ 6 ( t ) ℱ 6 ″ ( t ) − p + 1 2 ( ℱ 6 ′ ( t ) ) 2 ≥ ℱ 6 ( t ) ( ℱ 6 ″ ( t ) − 2 ( p + 1 ) ( ℱ 7 ( t ) + μ ) ) ≥ 2 ( p + 1 ) ℱ 6 ( t ) λ 1 ( p − 1 ) 2 ( p + 1 ) ‖ u 0 ‖ 2 2 − J ( u 0 ) − μ .

Pick a sufficiently small μ ∈ ( 0 , ζ ] that satisfies

λ 1 ( p − 1 ) 2 ( p + 1 ) ‖ u 0 ‖ 2 2 − J ( u 0 ) − μ ≥ 0 ,

where ζ ≔ λ 1 ( p − 1 ) 2 ( p + 1 ) ‖ u 0 ‖ 2 2 − J ( u 0 ) . Then, we have

ℱ 6 ( t ) ℱ 6 ″ ( t ) − p + 1 2 ( ℱ 6 ′ ( t ) ) 2 ≥ 0 ,

which means that the conditions appeared in Lemma 2.10 are met. At the same time, we verify that ℱ 6 ( 0 ) = T ‖ u 0 ‖ 2 2 + μ ν 2 > 0 , ℱ 6 ′ ( 0 ) ≥ 2 μ ν > 0 ; by Lemma 2.10, we derive

(8.10) T max ≤ 2 ℱ 6 ( 0 ) ( p − 1 ) ℱ 6 ′ ( 0 ) ≤ ‖ u 0 ‖ 2 2 ( p − 1 ) μ ν T + ν p − 1 .

Let ν be large enough such that

(8.11) ν ∈ ‖ u 0 ‖ 2 2 ( p − 1 ) μ , + ∞ .

Then, it follows from (8.10) that

(8.12) T max ≤ μ ν 2 ( p − 1 ) μ ν − ‖ u 0 ‖ 2 2 .

To estimate the upper-bound of blowup time, we describe a pair of ( μ , ν ) by the following set:

ℳ ≔ ( ν , μ ) ν ∈ ‖ u 0 ‖ 2 2 ( p − 1 ) ζ , + ∞ , μ ∈ ‖ u 0 ‖ 2 2 ( p − 1 ) ν , ζ .

Thus, the upper-bound of the blowup time can be displayed as:

(8.13) T max ≤ inf ( μ , ν ) ∈ ℳ μ ν 2 ( p − 1 ) μ ν − ‖ u 0 ‖ 2 2 .

Letting g ( μ , ν ) ≔ μ ν 2 ( p − 1 ) μ ν − ‖ u 0 ‖ 2 2 , and then differentiating g ( μ , ν ) with respect to μ , we deduce

g ′ ( μ , ν ) = − ν 2 ‖ u 0 ‖ 2 2 ( ( p − 1 ) μ ν − ‖ u 0 ‖ 2 2 ) 2 < 0 ,

which yields that g ( μ , ν ) is decreasing with respect to μ , Hence, for all ν , one can obtain

(8.14) inf ( μ , ν ) ∈ ℳ g ( μ , ν ) = inf ν g ( ζ , ν ) = inf ν ζ ν 2 ( p − 1 ) ζ ν − ‖ u 0 ‖ 2 2 .

By differentiating g ( ζ , ν ) with respect to ν and taking g ′ ( ζ , ν ) = 0 , we deduce the minimum point ν min = 2 ‖ u 0 ‖ 2 2 ( p − 1 ) ζ , then

inf ν g ( ζ , ν ) = g ( ζ , ν min ) = 4 ‖ u 0 ‖ 2 2 ( p − 1 ) ζ ,

which together with (8.14) implies that

(8.15) inf ( μ , ν ) ∈ ℳ g ( μ , ν ) = g ( ζ , ν min ) = 4 ‖ u 0 ‖ 2 2 ( p − 1 ) ζ .

Hence, from (8.13) and (8.15), we obtain

□ T max ≤ inf ( μ , ν ) ∈ ℳ g ( μ , ν ) = 4 ‖ u 0 ‖ 2 2 ( p − 1 ) ζ = 8 ( p + 1 ) ‖ u 0 ‖ 2 2 λ 1 ( p − 1 ) 2 ‖ u 0 ‖ 2 2 − 2 ( p 2 − 1 ) J ( u 0 ) .

In the following, we give a novel condition for obtaining decay or blowup of the solution under high initial energy for Problems (1.1)–(1.3).

Lemma 8.2

Let u ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) , then

  1. For any u ∈ N + , it follows that J ( u ) > 0 ;

  2. For every u ∈ N , we conclude J ( u ) = max λ ≥ 0 J ( λ u ) ;

  3. For arbitrary κ > 0 , the semiflow { u ∈ N + ∣ J ( u ) < κ , u ∈ ℋ 2 , 0 1 , N + 1 2 ( E ) } is bounded.

Proof

  1. For u ∈ N + , we have I ( u ) > 0 , which implies

    (8.16) J ( u ) = 1 2 ‖ ∇ E u ‖ 2 2 − 1 p + 1 ‖ u ‖ p + 1 p + 1 = p − 1 2 ( p + 1 ) ‖ ∇ E u ‖ 2 2 + 1 p + 1 I ( u ) > 0 .

  2. Due to u ∈ N , it is easy to see I ( u ) = 0 . Together with (iv) in Lemma 2.11, one can obtain

    d d λ J ( λ u ) = I ( λ u ) = 0 ,

    which yields λ = 1 and J ( u ) = max λ ≥ 0 J ( λ u ) for u ∈ N .

  3. According to J ( u ) < κ and I ( u ) > 0 , one obtains from (8.16) that ‖ ∇ E u ‖ 2 2 < κ 2 ( p + 1 ) p − 1 . Then the claim is proved.□

Proof of Theorem 3.7

Let u ( t ) ≔ S ( t ) u 0 for t ∈ [ 0 , T max ( u 0 ) ) . First, if u 0 ∈ N + satisfies ‖ u 0 ‖ 2 ≤ λ J ( u 0 ) , we assert that u ( t ) ∈ N + for any t ∈ [ 0 , T max ( u 0 ) ) . Assume by contradiction that there exists the first t 1 ∈ ( 0 , T max ( u 0 ) ) such that u ( t ) ∈ N + for 0 ≤ t < t 1 and u ( t 1 ) ∈ N ; together with (4.4) and (4.2), one deduces that

(8.17) ‖ u ( t 1 ) ‖ 2 2 < ‖ u 0 ‖ 2 2 ≤ λ J ( u 0 )

and

(8.18) J ( u ( t 1 ) ) < J ( u 0 ) .

As u ( t 1 ) ∈ N and (8.18), the definition of λ J ( u 0 ) gives ‖ u ( t 1 ) ‖ 2 2 ≥ λ J ( u 0 ) , which contradicts with (8.17); thus, u ( t ) ∈ N + . Together with (8.17) and condition (iii) in Lemma 8.2, { u ∈ N + ∣ J ( u ) < J ( u 0 ) } is bounded in ℋ 2 , 0 1 , N + 1 2 ( E ) for t ∈ [ 0 , T max ( u 0 ) ) such that T max ( u 0 ) = ∞ , i.e., u 0 ∈ G .

Additionally, it is easy to see from (4.4) and (4.2) for all w ∈ ω ( u 0 ) that ‖ w ‖ 2 2 < λ J ( u 0 ) and J ( w ) ≤ J ( u 0 ) . Obviously, it has been obtained earlier that ω ( u 0 ) ⊂ N + , which shows

(8.19) ω ( u 0 ) ∩ N = ∅ .

Because of the solution u ( t ) = S ( t ) u 0 for Problems (1.1)–(1.3) toward the stationary solution of equation (8.1) as t → ∞ , the Nehari flow N includes all the nontrivial solutions for Problems (8.1). Hence, (8.19) directly gives that ω ( u 0 ) = { 0 } , i.e. u 0 ∈ G 0 .

Finally, we will prove that if u 0 ∈ N − satisfies ‖ u 0 ‖ 2 2 ≥ Λ J ( u 0 ) , then T max ( u 0 ) < + ∞ . Similarly, we can construct a contradiction, which yields that u ( t ) ∈ N − for any t ∈ [ 0 , T max ( u 0 ) ) . Using the contrary assumption, T max ( u 0 ) = ∞ . Then, for every w ∈ ω ( u 0 ) , it follows that ‖ w ‖ 2 2 > Λ J ( u 0 ) and J ( w ) ≤ J ( u 0 ) via (4.4) and (4.2). Using a method similar to the previous one, we obtain from the definition of Λ J ( u 0 ) that ω ( u 0 ) ⊂ N − and ω ( u 0 ) ∩ N = ∅ . Because N includes all nontrivial solutions for Problems (8.1) and the solution u ( t ) converges to the solution of (8.1) as t → ∞ , the claim ω ( u 0 ) ∩ N = ∅ implies ω ( u 0 ) = { 0 } . But, the fact dist ( 0 , N − ) > 0 and ω ( u 0 ) ⊂ N − tell us that 0 ∉ ω ( u 0 ) . Finally, one can deduce ω ( u 0 ) = ∅ , which contradicts the assumption. Thus, T max ( u 0 ) < ∞ , i.e., u 0 ∈ ℬ .□

Subsequently, we estimate the lower-bound of blowup time and blowup rate when the blowup occurs.

Proof of Theorem 3.8

The lower-bound of blowup time. Let

(8.20) ℱ 8 ( t ) ≔ ∫ E w q ∣ u ∣ β d σ , t ∈ [ 0 , T max ) ,

where β is an odd number to be determined later. We determine that u is the solution of (1.1)–(1.3) that

d ℱ 8 ( t ) d t = β ( ∣ u ∣ β − 2 u , u t ) E = β ( ∣ u ∣ β − 2 u , Δ E u + ∣ u ∣ p − 1 u ) E = β − ( β − 1 ) ∫ E w q ∣ u ∣ β − 2 ∣ ∇ E u ∣ 2 d σ + ∫ E w q ∣ u ∣ β + p − 1 d σ .

From now, let v = ∣ u ∣ β 2 , then

(8.21) d ℱ 8 ( t ) d t = − 4 ( β − 1 ) β ∫ E w q ∣ ∇ E v ∣ 2 d σ + β ∫ E w q v 2 + 2 ( p − 1 ) β d σ .

To deal with the second term on the right-hand side of equation (8.21), one can use the edge-type Hölder inequality as follows:

(8.22) ∫ E w q v 2 + 2 ( p − 1 ) β d σ ≤ ∫ E w q v 4 d σ 1 3 ∫ E w q v 1 + 3 ( p − 1 ) β d σ 2 3 .

Then, applying the edge-type Poincaré inequality

(8.23) ∫ E w q v 4 d σ ≤ λ 1 − 2 ∫ E w q ∣ ∇ E v ∣ 4 d σ

and substituting (8.23) into (8.22), we see

(8.24) ∫ E w q v 2 + 2 ( p − 1 ) β d σ ≤ λ 1 − 2 3 ∫ E w q ∣ ∇ E v ∣ 2 d σ 2 3 ∫ E w q v 1 + 3 ( p − 1 ) β d σ 2 3 ,

where λ 1 is the optimal constant of the edge-type Ponicaré inequality.

Again from the edge-type Hölder inequality and v = ∣ u ∣ β 2 , we obtain

(8.25) ∫ E w q v 1 + 3 ( p − 1 ) β d σ = ∫ E w q ∣ u ∣ β + 3 ( p − 1 ) 2 d σ ≤ ∣ E ∣ 1 − ρ ( ℱ 8 ( t ) ) ρ ,

where ∣ E ∣ denotes the volume of E and ρ ≔ 1 2 + 3 ( p − 1 ) 2 β . Particularly, the inequality ρ < 1 naturally be ensured as long as the undetermined constant β satisfies β > 3 ( p − 1 ) . Thus, substituting (8.25) into (8.24), it follows that

(8.26) ∫ E w q v 2 + 2 ( p − 1 ) β d σ ≤ λ 1 − 2 3 ∣ E ∣ 2 3 ( 1 − ρ ) ∫ E w q ∣ ∇ E v ∣ 2 d σ 2 3 ( ℱ 8 ( t ) ) 2 ρ 3 .

Using the following Young’s inequality,

X r Y s ≤ r X + s Y , for r + s = 1 , X , Y ≥ 0 ,

Rewriting (8.26) with a parameter ε > 0 , it is easy to see

(8.27) ∫ E w q v 2 + 2 ( p − 1 ) β d σ ≤ λ 1 − 2 3 ∣ E ∣ 2 3 ( 1 − ρ ) ε ∫ E w q ∣ ∇ E v ∣ 2 d σ 2 3 ( ε − 2 ( ℱ 8 ( t ) ) 2 ρ ) 1 3 ≤ 2 3 λ 1 − 2 3 ∣ E ∣ 2 3 ( 1 − ρ ) ε ∫ E w q ∣ ∇ E v ∣ 2 d σ + 1 3 λ 1 − 2 3 ∣ E ∣ 2 3 ( 1 − ρ ) ε − 2 ( ℱ 8 ( t ) ) 2 ρ .

Combining (8.26) and (8.27) with (8.21), we obtain

d ℱ 8 ( t ) d t ≤ 2 ε 3 β λ 1 − 2 3 ∣ E ∣ 2 3 ( 1 − ρ ) − 4 ( β − 1 ) β ‖ ∇ E v ‖ 2 2 + β 3 ε 2 λ 1 − 2 3 ∣ E ∣ 2 3 ( 1 − ρ ) ( ℱ 8 ( t ) ) 2 ρ .

Choosing ε to make the coefficient of ‖ ∇ E v ‖ 2 2 vanish, we reach to

(8.28) d ℱ 8 ( t ) d t ≤ γ ( ℱ 8 ( t ) ) 2 ρ ,

where γ = β 3 ε 2 λ 1 − 2 3 ∣ E ∣ 2 3 ( 1 − ρ ) . Or upon integration, we have for t < T max :

1 ℱ 8 ( 0 ) 2 ρ − 1 − 1 ℱ 8 ( t ) 2 ρ − 1 ≤ γ ( 2 ρ − 1 ) t .

So, letting t → T max , we arrive at

T max ≥ 1 γ ( 2 ρ − 1 ) ( ℱ 8 ( 0 ) ) 2 ρ − 1 = 36 λ 1 ( β − 1 ) 2 β 3 ( p − 1 ) ∣ E ∣ 1 − 3 ( p − 1 ) β ‖ u 0 ‖ β 3 ( p − 1 ) .

The lower-bound of blowup rate. Moreover, integrating (8.28) from t to T max , we obtain

(8.29) T max − t ≥ ∫ ℱ 8 ( t ) ∞ d θ γ θ 2 ρ ≔ F ( ℱ 8 ( t ) ) .

Obviously, as a decreasing function of ℱ 8 ( t ) , the function F ( ℱ 8 ( t ) ) exists inverse function F − 1 , which is still decreasing. Thus, one claims ℱ 8 ( t ) ≥ F − 1 ( T max − t ) , which yields that the lower-bound of blowup rate existed. By calculating the generalized integral in (8.29), it follows that

T max − t ≥ 1 γ ( 2 ρ − 1 ) ( ℱ 8 ( t ) ) 1 − 2 ρ ,

which implies

ℱ 8 ( t ) ≥ ( γ ( 2 ρ − 1 ) ( T max − t ) ) 1 1 − 2 ρ .

It can also be more clearly expressed as:

□ ‖ u ( t ) ‖ β ≥ β 4 ∣ E ∣ 1 − 3 ( p − 1 ) β 36 ( β − 1 ) 2 λ 2 1 β ( 1 − 2 ρ ) ( T max − t ) 1 β ( 1 − 2 ρ ) .

  1. Funding information: This work was supported by the Heilongjiang Provincial Natural Science Foundation of China (LH2021A001) and Fundamental Research Funds in Heilongjiang Provincial Universities of China (2022-KYYWF-1112).

  2. Conflict of interest: The author declares that there is no conflict of interest.

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Received: 2023-03-06
Revised: 2023-07-11
Accepted: 2023-10-27
Published Online: 2023-11-28

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  35. Approximations of center manifolds for delay stochastic differential equations with additive noise
  36. Periodic solutions to a class of distributed delay differential equations via variational methods
  37. Groundstate for the Schrödinger-Poisson-Slater equation involving the Coulomb-Sobolev critical exponent
  38. Existence of nontrivial solutions for the Klein-Gordon-Maxwell system with Berestycki-Lions conditions
  39. Global Sobolev regular solution for Boussinesq system
  40. Normalized solutions for the p-Laplacian equation with a trapping potential
  41. Nonlinear elliptic–parabolic problem involving p-Dirichlet-to-Neumann operator with critical exponent
  42. Blow-up for compressible Euler system with space-dependent damping in 1-D
  43. High energy solutions of general Kirchhoff type equations without the Ambrosetti-Rabinowitz type condition
  44. On the dynamics of grounded shallow ice sheets: Modeling and analysis
  45. A survey on some vanishing viscosity limit results
  46. Blow-up for logarithmic viscoelastic equations with delay and acoustic boundary conditions
  47. Generalized Liouville theorem for viscosity solutions to a singular Monge-Ampère equation
  48. Front propagation in a double degenerate equation with delay
  49. Positive solutions for a class of singular (p, q)-equations
  50. Higher integrability for anisotropic parabolic systems of p-Laplace type
  51. The Poincaré map of degenerate monodromic singularities with Puiseux inverse integrating factor
  52. On a system of multi-component Ginzburg-Landau vortices
  53. Existence and concentration of solutions to Kirchhoff-type equations in ℝ2 with steep potential well vanishing at infinity and exponential critical nonlinearities
  54. Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities
  55. On double phase Kirchhoff problems with singular nonlinearity
  56. Estimates for eigenvalues of the Neumann and Steklov problems
  57. Nodal solutions with a prescribed number of nodes for the Kirchhoff-type problem with an asymptotically cubic term
  58. Dirichlet problems involving the Hardy-Leray operators with multiple polars
  59. Incompressible limit for compressible viscoelastic flows with large velocity
  60. Characterizations for the genuine Calderón-Zygmund operators and commutators on generalized Orlicz-Morrey spaces
  61. Non-local gradients in bounded domains motivated by continuum mechanics: Fundamental theorem of calculus and embeddings
  62. Noncoercive parabolic obstacle problems
  63. Touchdown solutions in general MEMS models
  64. Existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential
  65. Short-time existence of a quasi-stationary fluid–structure interaction problem for plaque growth
  66. Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs
  67. Symmetries of Ricci flows
  68. Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type
  69. On the topological gradient method for an inverse problem resolution
  70. Supersolutions to nonautonomous Choquard equations in general domains
  71. Uniform complex time heat Kernel estimates without Gaussian bounds
  72. Global existence for time-dependent damped wave equations with nonlinear memory
  73. Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation
  74. Reversed Hardy-Littlewood-Sobolev inequalities with weights on the Heisenberg group
  75. Lamé system with weak damping and nonlinear time-varying delay
  76. Global well posedness for the semilinear edge-degenerate parabolic equations on singular manifolds
  77. Blowup in L1(Ω)-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms
  78. Boundary regularity results for minimisers of convex functionals with (p, q)-growth
  79. Parametric singular double phase Dirichlet problems
  80. Special Issue on Nonlinear analysis: Perspectives and synergies
  81. Editorial to Special issue “Nonlinear analysis: Perspectives and synergies”
  82. Identification of discontinuous parameters in double phase obstacle problems
  83. Global boundedness to a 3D chemotaxis-Stokes system with porous medium cell diffusion and general sensitivity
  84. On regular solutions to compressible radiation hydrodynamic equations with far field vacuum
  85. On Cauchy problem for fractional parabolic-elliptic Keller-Segel model
  86. The evolution of immersed locally convex plane curves driven by anisotropic curvature flow
  87. Stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data
  88. On concave perturbations of a periodic elliptic problem in R2 involving critical exponential growth
  89. Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition
https://doi.org/10.1515/anona-2023-0117
Keywords for this article
global existence; finite-time blowup; nonlinear parabolic equation; singular manifolds
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  34. Fujita-type theorems for a quasilinear parabolic differential inequality with weighted nonlocal source term
  35. Approximations of center manifolds for delay stochastic differential equations with additive noise
  36. Periodic solutions to a class of distributed delay differential equations via variational methods
  37. Groundstate for the Schrödinger-Poisson-Slater equation involving the Coulomb-Sobolev critical exponent
  38. Existence of nontrivial solutions for the Klein-Gordon-Maxwell system with Berestycki-Lions conditions
  39. Global Sobolev regular solution for Boussinesq system
  40. Normalized solutions for the p-Laplacian equation with a trapping potential
  41. Nonlinear elliptic–parabolic problem involving p-Dirichlet-to-Neumann operator with critical exponent
  42. Blow-up for compressible Euler system with space-dependent damping in 1-D
  43. High energy solutions of general Kirchhoff type equations without the Ambrosetti-Rabinowitz type condition
  44. On the dynamics of grounded shallow ice sheets: Modeling and analysis
  45. A survey on some vanishing viscosity limit results
  46. Blow-up for logarithmic viscoelastic equations with delay and acoustic boundary conditions
  47. Generalized Liouville theorem for viscosity solutions to a singular Monge-Ampère equation
  48. Front propagation in a double degenerate equation with delay
  49. Positive solutions for a class of singular (p, q)-equations
  50. Higher integrability for anisotropic parabolic systems of p-Laplace type
  51. The Poincaré map of degenerate monodromic singularities with Puiseux inverse integrating factor
  52. On a system of multi-component Ginzburg-Landau vortices
  53. Existence and concentration of solutions to Kirchhoff-type equations in ℝ2 with steep potential well vanishing at infinity and exponential critical nonlinearities
  54. Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities
  55. On double phase Kirchhoff problems with singular nonlinearity
  56. Estimates for eigenvalues of the Neumann and Steklov problems
  57. Nodal solutions with a prescribed number of nodes for the Kirchhoff-type problem with an asymptotically cubic term
  58. Dirichlet problems involving the Hardy-Leray operators with multiple polars
  59. Incompressible limit for compressible viscoelastic flows with large velocity
  60. Characterizations for the genuine Calderón-Zygmund operators and commutators on generalized Orlicz-Morrey spaces
  61. Non-local gradients in bounded domains motivated by continuum mechanics: Fundamental theorem of calculus and embeddings
  62. Noncoercive parabolic obstacle problems
  63. Touchdown solutions in general MEMS models
  64. Existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential
  65. Short-time existence of a quasi-stationary fluid–structure interaction problem for plaque growth
  66. Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs
  67. Symmetries of Ricci flows
  68. Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type
  69. On the topological gradient method for an inverse problem resolution
  70. Supersolutions to nonautonomous Choquard equations in general domains
  71. Uniform complex time heat Kernel estimates without Gaussian bounds
  72. Global existence for time-dependent damped wave equations with nonlinear memory
  73. Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation
  74. Reversed Hardy-Littlewood-Sobolev inequalities with weights on the Heisenberg group
  75. Lamé system with weak damping and nonlinear time-varying delay
  76. Global well posedness for the semilinear edge-degenerate parabolic equations on singular manifolds
  77. Blowup in L1(Ω)-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms
  78. Boundary regularity results for minimisers of convex functionals with (p, q)-growth
  79. Parametric singular double phase Dirichlet problems
  80. Special Issue on Nonlinear analysis: Perspectives and synergies
  81. Editorial to Special issue “Nonlinear analysis: Perspectives and synergies”
  82. Identification of discontinuous parameters in double phase obstacle problems
  83. Global boundedness to a 3D chemotaxis-Stokes system with porous medium cell diffusion and general sensitivity
  84. On regular solutions to compressible radiation hydrodynamic equations with far field vacuum
  85. On Cauchy problem for fractional parabolic-elliptic Keller-Segel model
  86. The evolution of immersed locally convex plane curves driven by anisotropic curvature flow
  87. Stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data
  88. On concave perturbations of a periodic elliptic problem in R2 involving critical exponential growth
  89. Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition
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