On the weakly degenerate Allen-Cahn equation
-
Maicon Sônego
Abstract
In this paper we consider a one-dimensional Allen-Cahn equation with degeneracy in the interior of the domain and Neumann boundary conditions. We allow the diffusivity coefficient vanish at some point of the space domain and we are addressed on the existence of stable non-constant solution.
1 Introduction
Consider the following semi-linear problem
where f(u) = u − u3 and a(⋅) ∈ C([0, 1]) satisfies
a(x0) = 0 at some x0 ∈ (0, 1);
a(x) > 0 for x ∈ [0, 1] ∖ {x0} and
1/a ∈ L1(0, 1).
With these conditions, the parabolic problem (1.1) is called weakly degenerate (see [1], for instance) with degeneracy in the interior of the space domain. A typical example for a(⋅) satisfying the conditions (H1)–(H3) is given by a(x) = ∣x − x0∣α (0 < α < 1).
In this paper we are concerned to ensure the existence of solution to the evolution problem (1.1) and, mainly, to study the existence of stable solutions of the variational problem where the critical points of energy functional are stationary solutions of (1.1); that is
Roughly speaking, we study the role of the degeneracy of function a(⋅) at x0 to the existence of a local minimizer of the energy functional associated to (1.2). This class of solutions (which we call stable solutions, see Definition 3.1 and Remark 3.2), in general, enjoys better qualitative and quantitative properties (monotonicity and symmetry, for instance) than the other solutions. In particular, when a minimizer is isolated – in this case we say asymptotically stable – it can describe the whole dynamics of the corresponding parabolic problem. We refer to the excellent monograph [2] for a comprehensive and complete presentation of the main results available on stable solutions.
Degenerate problems have always attracted the attention of many authors, [1, 3, 4, 5, 6, 7] and references therein. In these works it is assumed that the function a(⋅) degenerates at the boundary or in the interior of the space domain and the results are mainly related to the theory of control. In particular, in [1] the authors study degenerate parabolic problems with interior degeneracy, under Dirichlet boundary conditions, and show that under suitable assumptions, they generate analytic semi-groups. In addition, some applications to linear and semi-linear parabolic problems are provided. Similar results were achieved in [7], under Neumann boundary conditions, whose focus was to obtain Carleman estimates. With regard to existence of solution to the problem (1.1), the main results of the present work are based on these last two articles.
Here we will focus on a particular reaction term related to Allen-Cahn problem, f(u) = u − u3. The Allen-Cahn equation has its origin in the theory of phase transitions ([8]) and it is used as a model for some nonlinear reaction-diffusion processes. For instance, assume that there are two populations 𝓐 and 𝓑 and that u is a density measuring the percentage of the two populations at every point; that is, if u(x) = 1 (u(x) = −1) at a point x, we have only population 𝓐 (population 𝓑) at x and u(x) = 0 means that at x we have 50% of 𝓐 and 50% of 𝓑. The non-homogeneity of the medium is expressed by the space dependence of the diffusion coefficient a(⋅). If this coefficient vanishes at some point, then this will lead to the interruption of the migration and/or interaction of the species. Evidently, u ≡ 1 or u ≡ −1 are two stable states of the system, however, our goal is to obtain existence of stable non-constant solutions.
The study of existence or non-existence of stable solutions to semi-linear problems (in particular those of the Allen-Cahn type), with a(x) ≡ 1 or a(⋅) strictly positive, is the subject of numerous articles and books. In the face of an extensive literature, we cite [9, 10, 11, 12, 13, 14]and references therein. Of course, allowing a(⋅) vanishes at some point brings with it several technical difficulties. Since the operator Au := (aux)x is no longer elliptic (sometimes called degenerate elliptic), some basic analysis tools – such as the Maximum Principles, Hopf’s Lemma and Spectral Theory – can not be used. To the best of our knowledge, the present work is the first to study the role of a degenerate diffusion coefficient related to the existence of stable solutions.
The technique we use here stems from the one presented in [11] (see also [13, 14, 15]). After finding a invariant set for the flux of (1.1) – hence the need to ensure the existence of solution – we use some variational techniques, in a convenient weighted Sobolev space, to show existence of a local minimizer of the energy functional in this invariant set.
Finally, let us remark that even our result of existence of solution to the problem (1.1), based on the works [1, 7], it is new and although it is essential to our main result, it may have interest of its own.
2 Existence of solution
In this section we deal with the existence of solution to the problem (1.1). It is important to note that the results of this section can easily be adapted to more general semi-linear equations.
Firstly, we should note that Au = (aux)x, in a suitable domain, generates a analytic semigroup. For this purpose, we introduce the following weighted spaces (sometimes, we use ′ to denote the derivative with respect to x):
with the norm
and
with
Definition 2.1
If u0 ∈ L2(0, 1), a function u is said to be a weak solution of (1.1) if
and
for all ϕ ∈ H1(0, T; L2(0, 1)) ∩ L2(0, T;
Now we define the operator A by D(A) := {u ∈
The proof of the next two results can be found in [1, Lemma 2.1] (see also [7, Lemma 2.1]) and [7, Theorem 2.1], respectively.
Lemma 2.2
For all (u, v) ∈ D(A) ×
Theorem 2.3
The operator A : D(A) → L2(0, 1) is self-adjoint, nonpositive on L2(0, 1) and it generates an analytic contraction semigroup.
Now we proceed as in [1]. Since A is a generator, and setting B(t)u := u, working in the spaces considered above, we can prove that the problem below (with c ∈ L∞(ℝ+ × (0, 1))) is well-posed in the sense of semigroup theory using some well-known perturbation technique (see [16], for instance).
Hence, for a fixed T > 0 we get the following result.
Theorem 2.4
If c(⋅, x) ∈ C1(ℝ+) for all x ∈ [0, 1] and u0 ∈ D(A) then there is a unique weak solution
of (2.4) and
for a positive constant C.
Next result can be found in [1, Theorem 5.4]
Lemma 2.5
The set H1(0, T; L2(0, 1)) ∩ L2(0, T;
We are now in position to state the main result of this section. The proof follows the steps of [1, Theorem 4.12], however, some modifications are necessary because we consider Neumann boundary conditions and the specific nonlinear term f(u) = u − u3.
Proof
We set X := C([0, T]; L2(0, 1)) ∩ L2(0, T;
where uv is the unique solution of
We use Theorem 2.4 to ensure that (2.7) has a unique weak solution u ∈ X. Now, we will prove that 𝓣 has a fixed point uv (that is, 𝓣(uv) = uv) to conclude that uv is a solution of (1.1).
By Schauder’s Theorem, it is sufficient to prove that
𝓣:BX → BX,
𝓣 is a compact function and
𝓣 is a continuous function,
where
CT it is the same constant of Theorem 2.4 and
The items (i) and (ii) are consequence of Theorem 2.4 and Lemma 2.5, respectively.
To prove (iii) we take vk ∈ X such that vk → v in X, as k → ∞. We will prove that 𝓣(vk) = uvk := uk → 𝓣(v) = uv in X, as k → ∞. Recall that uk and uv are the solutions of (2.7) associated to vk and v, respectively. As D(A) is dense in
Multiplying the equation
by a test function ϕ ∈ H1(0, T; L2(0, 1)) ∩ L2(0, T;
We recall that
Since uk converges strongly to ū in X, it is immediate to prove (a)-(c). In order to prove (d) we recall that cvk(t, x) = 1 − (vk)2(t, x) and vk converges strongly to v in X. Therefore, we can conclude that vk converges to v a.e. as well as uk converges to ū a.e.. Thus,
and (d) holds by an application of Lebesgue Theorem.
We proved that ū is the unique weak solution of (2.4) in Y ⊂ X associated to v; that is ū = uv and (iii) is proved. It follows that 𝓣 has a fixed point uv ∈ Y which is a solution of (1.1). The theorem is proved.□
3 Existence of stable solutions
We start by defining an energy functional E :
where
It is not difficult to verify that E is twice continuously differentiable and a simple computation give us that its critical points are weak stationary solutions of (1.1) (i.e. weak solutions of (1.2)).
Definition 3.1
Let u be a weak solution to (1.2). We say that u is stable if
for any ϕ ∈
Remark 3.2
Note that (3.2) is equivalent to the second variation of the energy functional E(⋅) at u to be non-negative. Therefore, local minimizers of E are stable solutions of (1.2).
In order to state our next results, we set Il, Ir two sub-intervals of (0, 1) such that Il ⊂ (0,x0) and Ir ⊂ (x0, 1).
Lemma 3.3
(Poincaré-type inequality). There exists a constant 𝓒j (j = l, r), depending only on Ij and a∣Ij, such that
for each function u ∈
Proof
For brevity, we omit the sub-indices j = l, r. We argue by contradiction; that is, we suppose that for each k ∈ ℕ, there exists uk ∈
where
We renormalize by defining
It follows that
In particular the functions {vk}k∈ℕ are bounded in
From (3.5)
If ϕ ∈
Hence v ∈ H1(I) and v′ = 0 a.e. in I. It follows that v is constant in I which is a contradiction with (3.8).□
Now, we set the positive number
where 𝓒j (j = l, r) is the optimal constant in (3.3).
Our main result is stated as follow.
Theorem 3.4
If there is δ > 0 such that δ < 2ϵ0, Qδ := [x0 − δ, x0+δ] ∩ (Il ∪ Ir) = ∅ and
then (1.2) admits a non-constant stable solution.
For each t > 0, we consider
defined by
where u(t, x) is the solution of (1.1) with u(0, x) = u0(x) given by Theorem 2.6. Then t → T(t)[u0(x)] is continuous for any u0 ∈
Proposition 3.5
Consider the set
If u0 ∈ Λ then T(t)u0 ∈ Λ for all t > 0.
Proof
−1 ≤ T(t)u0 = u(t, x) ≤ 1 in [0, 1] for all t > 0.
If there is (t̄, x̄) such that u(t̄, x̄) > 1 then if T > t̄
and uM = u(t̃, x̃) at some (t̃, x̃) ∈ (0, T] × [0, 1]. We have three possibilities:
x̃ ∈ (0, 1) ∖ {x0},
x̃ = x0 or
x̃ ∈ {0, 1}.
If (i) holds we can apply the Maximum Principle in a sub-interval that does not contain x0. Hence, we have a contradiction since u(0, x) = u0(x) ≤ 1. For (iii), we use a one-dimensional version of Hopf’s Lemma and that ux(t̃,0) = ux(t̃, 1) = 0 to get a contradiction. Finally, we note that
and then ut(t,x0) < 0 for t near of t̃. It follows that (ii) does not occur. Therefore u(t, x) ≤ 1 (t > 0) and analogously we prove −1 ≤ u(t, x) for all t > 0 which proves the Claim 1.
E(u(t, x)) <
Indeed, it is true because
∫Ilu(t, x)dx < 0 and ∫Iru(t, x)dx > 0 for all t > 0.
By contradiction, let t1 > 0 be such that u1(x) := u(t1, x) satisfies
Then, by Lemma 3.3
Now, note that
Hence,
and
We also have
Therefore, as F(u1) ≤ F(1) = 1/4
and
if
if
In both cases we have a contradiction. Similarly we prove that ∫Iru(t, x)dx > 0 for all t > 0. It is proved that Λ is invariant under T(t) for t ≥ 0.□
Proposition 3.6
If Λ ≠ ∅ then (1.1) has at least one non-constant stationary solution u ∈ Λ which is stable in
Proof
If v ∈ Λ, by Proposition 3.5, γ(v) := {T(t)v; t ≥ 0} ⊂ Λ and because the system is gradient (E is a functional of Lyapunov) γ(v) is compact. It follows that
If 𝓔 is the set of all equilibrium solutions to (1.1) then ω(v) ⊂ 𝓔. Hence, if u ∈ ω(v) then −1 ≤ u ≤ 1, E(u) ≤ E(v) < ϵ0 − (1/4) and, as before, it is possible to prove that ∫Il u < 0, ∫Ir u > 0; that is, ω(v) ⊂ Λ.
Therefore, if v ∈ Λ, Λ ∩ 𝓔 ≠ ∅. Moreover Λ ∩ 𝓔 is compact in
It is not difficult to see that 𝓔 is closed in
Because of the continuity of E, there is e0 ∈ Λ ∩ 𝓔 such that E(e0) ≤ E(v) for all v ∈ Λ ∩ 𝓔. More than that, E(e0) ≤ E(v) for all v ∈ Λ since otherwise there would be v1 ∈ Λ such that E(v1) < E(e0). As before ω(v1) ⊂ Λ and then, for all v ∈ ω(v1),
which is a contradiction.
The next step is to prove that e0 is a local minimum of E in
Λ1 := {u ∈
Λ2 := {u ∈
Λ3 := {u ∈
Λ4 := {u ∈
Indeed, Λj (j = 2, …, 4) are open in
e0 ∈
which contradicts E(e0) ≤ E(v) for all v ∈ Λ. Therefore e0 ∈ Λ4 and this proves Claim 2. Thus, e0 is a local minimum of E in
i.e.
which proves that e0 is a stable non-constant (e0 ∈ Λ) solution of (1.2).□
Finally, we are in position to prove our main result.
Proof of the Theorem 3.4
We shall prove that Λ ≠ ∅ and the theorem follows by Proposition 3.6.
Consider the signed distance function defined in ℝ by
and ξ : ℝ → ℝ defined by
We will show that w0(x) := ξ(d(x, x0))∣(0,1) ∈ Λ. It is not difficult to see that w0 ∈
Now, if we set
By (3.10), E(w0) < ϵ0 − (1/4) and then w0 ∈ Λ which proves the Theorem 3.4.□
Remark 3.7
It is common to say that the solution e0 obtained in this work “is trapped at the bottom of an energy well”. That is why we say that it is stable. A natural question is whether such a solution is asymptotically stable; that is, to know if the solutions of the corresponding parabolic problem (problem (1.1)) with the initial data near e0 tend to e0 as t → ∞. This type of stability – also called linearized stability or Lyapunov stability – in general, it is accomplished by studying the spectrum of the corresponding linearized problem. However, the degeneracy considered here makes it impossible to carry out such a study. This question has been studied when the degenerate operator has an uniformly elliptic direction; which, obviously, the operator considered here does not have. For all the details on this issue, we cite [17].
Remark 3.8
In the case where A = (aux)x is strongly degenerate; that is, a ∈ W1,∞(0, 1) satisfies (H1) and (H2) but 1/a ∉ L1(0, 1), can not be carried out in an analogous way. The hypotheses (H1)-(H3) are essential in all results of this work which makes the strongly degenerate case an open problem.
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© 2020 M. Sônego, published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 Public License.
Articles in the same Issue
- Frontmatter
- On the moving plane method for boundary blow-up solutions to semilinear elliptic equations
- Regularity of solutions of the parabolic normalized p-Laplace equation
- Cahn–Hilliard equation on the boundary with bulk condition of Allen–Cahn type
- Blow-up solutions for fully nonlinear equations: Existence, asymptotic estimates and uniqueness
- Radon measure-valued solutions of first order scalar conservation laws
- Ground state solutions for a semilinear elliptic problem with critical-subcritical growth
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- Remarks on a nonlinear nonlocal operator in Orlicz spaces
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Articles in the same Issue
- Frontmatter
- On the moving plane method for boundary blow-up solutions to semilinear elliptic equations
- Regularity of solutions of the parabolic normalized p-Laplace equation
- Cahn–Hilliard equation on the boundary with bulk condition of Allen–Cahn type
- Blow-up solutions for fully nonlinear equations: Existence, asymptotic estimates and uniqueness
- Radon measure-valued solutions of first order scalar conservation laws
- Ground state solutions for a semilinear elliptic problem with critical-subcritical growth
- Generalized solutions of variational problems and applications
- Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity
- Nonlinear Sherman-type inequalities
- Global regularity for systems with p-structure depending on the symmetric gradient
- Homogenization of a net of periodic critically scaled boundary obstacles related to reverse osmosis “nano-composite” membranes
- Noncoercive resonant (p,2)-equations with concave terms
- Evolutionary quasi-variational and variational inequalities with constraints on the derivatives
- Sharp estimates on the first Dirichlet eigenvalue of nonlinear elliptic operators via maximum principle
- Localization and multiplicity in the homogenization of nonlinear problems
- Remarks on a nonlinear nonlocal operator in Orlicz spaces
- A Picone identity for variable exponent operators and applications
- On the weakly degenerate Allen-Cahn equation
- Continuity results for parametric nonlinear singular Dirichlet problems
- Construction of type I blowup solutions for a higher order semilinear parabolic equation
- Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions
- Comparison results for nonlinear divergence structure elliptic PDE’s
- Constant sign and nodal solutions for parametric (p, 2)-equations
- Monotonicity formulas for coupled elliptic gradient systems with applications
- Berestycki-Lions conditions on ground state solutions for a Nonlinear Schrödinger equation with variable potentials
- A class of semipositone p-Laplacian problems with a critical growth reaction term
- The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in L1
- Reconstruction of Tesla micro-valve using topological sensitivity analysis
- Lewy-Stampacchia’s inequality for a pseudomonotone parabolic problem
- Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term
- Regularity Criteria for Navier-Stokes Equations with Slip Boundary Conditions on Non-flat Boundaries via Two Velocity Components
- Homoclinics for singular strong force Lagrangian systems
- A constructive method for convex solutions of a class of nonlinear Black-Scholes equations
- On a class of nonlocal nonlinear Schrödinger equations with potential well
- Superlinear Schrödinger–Kirchhoff type problems involving the fractional p–Laplacian and critical exponent
- Regularity for minimizers for functionals of double phase with variable exponents
- Boundary blow-up solutions to the Monge-Ampère equation: Sharp conditions and asymptotic behavior
- Homogenisation with error estimates of attractors for damped semi-linear anisotropic wave equations
- A-priori bounds for quasilinear problems in critical dimension
- Critical growth elliptic problems involving Hardy-Littlewood-Sobolev critical exponent in non-contractible domains
- On the Sobolev space of functions with derivative of logarithmic order
- On a logarithmic Hartree equation
- Critical elliptic systems involving multiple strongly–coupled Hardy–type terms
- Sharp conditions of global existence for nonlinear Schrödinger equation with a harmonic potential
- Existence for (p, q) critical systems in the Heisenberg group
- Periodic traveling fronts for partially degenerate reaction-diffusion systems with bistable and time-periodic nonlinearity
- Some hemivariational inequalities in the Euclidean space
- Existence of standing waves for quasi-linear Schrödinger equations on Tn
- Periodic solutions for second order differential equations with indefinite singularities
- On the Hölder continuity for a class of vectorial problems
- Bifurcations of nontrivial solutions of a cubic Helmholtz system
- On the exact multiplicity of stable ground states of non-Lipschitz semilinear elliptic equations for some classes of starshaped sets
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- Positive solutions for diffusive Logistic equation with refuge
- Null controllability for a degenerate population model in divergence form via Carleman estimates
- Eigenvalues for a class of singular problems involving p(x)-Biharmonic operator and q(x)-Hardy potential
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- Solvability of an infinite system of nonlinear integral equations of Volterra-Hammerstein type
- The superposition operator in the space of functions continuous and converging at infinity on the real half-axis
- Estimates by gap potentials of free homotopy decompositions of critical Sobolev maps
- Pseudo almost periodic solutions for a class of differential equation with delays depending on state
- Normalized multi-bump solutions for saturable Schrödinger equations
- Some inequalities and superposition operator in the space of regulated functions
- Area Integral Characterization of Hardy space H1L related to Degenerate Schrödinger Operators
- Bifurcation of time-periodic solutions for the incompressible flow of nematic liquid crystals in three dimension
- Morrey estimates for a class of elliptic equations with drift term
- A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems
- Global and non global solutions for a class of coupled parabolic systems
- On the analysis of a geometrically selective turbulence model
- Multiplicity of positive solutions for quasilinear elliptic equations involving critical nonlinearity
- Lack of smoothing for bounded solutions of a semilinear parabolic equation
- Gradient estimates for the fundamental solution of Lévy type operator
- π/4-tangentiality of solutions for one-dimensional Minkowski-curvature problems
- On the existence and multiplicity of solutions to fractional Lane-Emden elliptic systems involving measures
- Anisotropic problems with unbalanced growth
- On a fractional thin film equation
- Minimum action solutions of nonhomogeneous Schrödinger equations
- Global existence and blow-up of weak solutions for a class of fractional p-Laplacian evolution equations
- Optimal rearrangement problem and normalized obstacle problem in the fractional setting
- A few problems connected with invariant measures of Markov maps - verification of some claims and opinions that circulate in the literature
