Well-posedness of damped Kirchhoff-type wave equation with fractional Laplacian
-
Shaohua Chen
and
Meina Zhang
Abstract
In the present paper, we study the well-posedness of the solution to the initial boundary value problem for the damped Kirchhoff-type wave equation with fractional Laplacian. First, the existence and uniqueness of the local solution are established by the Banach fixed point theorem. Then, the global existence and finite time blowup of the solution are derived at the subcritical and critical initial energy levels. Finally, the finite time blowup of the solution and upper bound and lower bound estimate of blowup time are given at the arbitrarily positive initial energy level.
1 Introduction
We consider the initial boundary value problem (IBVP) of the damped Kirchhoff-type wave equation with fractional Laplacian
where
and the power q of the weak damping and the dimension N satisfy
Combining (1.2) and (1.3) ensures
The fractional Laplacian operator is defined by
for any
which satisfies
M(m) is an increasing and continuous function;
M(m) is continuously differentiable for all
In problem (1.1), the Gagliardo seminorm of u is defined by
where
The Kirchhoff problems can be used to describe the transverse oscillations of a stretched string with nonlocal flexural rigidity, which continuously depend on the Sobolev deflection norm of the displacement u via the Kirchhoff function M [1], [2]. The original version of the equation in (1.1) is
where t ≥ 0, 0 < x < L, u(x, t) is the lateral displacement at the space x and time t, E is the Young modulus, h is the cross-section area, ρ is the mass density, L is the length, P 0 is the initial axial tension, δ is the resistance modulus, and f is the external force. Equation (1.6) with f = 0 was first presented by Kirchhoff in [3], which is an extension of the classical D’Alembert wave equation considering the changing effects in the length of the string during the vibrations, then Kirchhoff in [4] introduced equation (1.6) with δ = f = 0. Ebihara et al. in [5] studied (1.6) with P 0 = δ = f = 0 and obtained the existence and uniqueness of the local solution to the corresponding IBVP.
To systematically recall the related work of problem (1.1), we give the following general form of the Kirchhoff-type wave equation
For g = f = 0 and M is non-degenerate, D’Ancona and Spagnolo in [1] proved the global existence of the solution to the Cauchy problem of (1.7) in the analytic periodic case. Lasiecka and Ong in [6] obtained the global existence, uniqueness and uniform decay rate of the solution to the IBVP of (1.7). Miranda and Jutuca in [7] established the global existence and the boundary stability to the initial-inhomogeneous boundary value problem of (1.7). Baldi and Haus in [8] gave a lower bound of the existence time of the solution to the IBVP of (1.7) with periodic boundary condition. For g = u
t
, f = 0 and M is non-degenerate, Vicente in [9] proved the well-posedness and exponential stability of the energy to the IBVP of (1.7) with non-porous acoustic boundary condition. For g = u
t
, f = |u|
α
u, α > 0 and M is mildly degenerate, Ono in [10] studied the constraint mechanism between damping and nonlinear source term and obtained the global existence and finite time blowup of the solution to the IBVP of (1.7) in the framework of the potential well theory. For
It is worthy mentioning that many mathematical models involving fractional operators have attracted lots of attention. The fractional partial differential equations arise in a quite natural way in many applications, such as in the fields of physics, chemistry, biology and finance [22]. In Applebaum [23], the fractional Laplacian is viewed as the infinitesimal generator of stable radially symmetric Lévy processes. We point out that (−Δ)
s
can be reduced to the classical Laplace operator −Δ as s → 1, and we can see Proposition 4.4 in Nezza et al. [24] for more details. In particular, thanks to the pioneering work of Caffarelli and Silvestre [25], many interesting results in some classic elliptic problems have been extended in the fractional Laplacian setting [26], [27]. Following Caffarelli and Roquejoffre [28], Fiscella and Valdinoci in [29] proposed a stationary Kirchhoff variational model in bounded regular domains of
where M(m) = m
γ−1,
For the investigation of the Kirchhoff problem with the classical Laplacian, there have been plentiful related results, and only a few works are concerned with the fractional Laplacian case, especially both the Kirchhoff term and strong damping with fractional Laplacian. The significant feature of the fractional Laplacian is the nonlocal property, which leads to more complicated nonlinearity for the Kirchhoff term. From the perspective of the energy of the system, the dampings and nonlinear source term are the factors causing decay and nonlinear growth of the energy, respectively, which lead to global existence and blowup of the solution, respectively also. The strong damping with fractional Laplacian, nonlinear weak damping and nonlinear source term in (1.1) not only bring substantial difficulties to constructing the variational structure but also to analyzing the constraint relationship between dampings and nonlinear source term. Inspired by these challenges and the works we recalled, in the present paper, we shall systematically study the well-posedness of solutions to problem (1.1) at three initial energy levels, i.e., subcritical, critical and arbitrarily positive initial energy levels, and further prove the asymptotic behavior of the global solution and estimate the blowup time of the finite time blowup solution. Since we consider two types of dampings, i.e., nonlinear weak damping and strong damping, especially the strong damping is nonlocal, the standard concavity method is ineffective, which means we need propose new strategies to construct appropriate auxiliary functional to establish corresponding concave inequalities for problem (1.1).
The rest of the paper is organized as follows. In Section 2, we give some notations and recall necessary definitions and properties of the fractional Sobolev spaces and potential well theory. In Section 3, we establish the local existence of the solution by Banach fixed point theorem. In Section 4 and Section 5, we prove the global existence, asymptotic behavior and finite time blowup of the solution to problem (1.1) at the subcritical initial energy level, i.e., E(0) < d and the critical initial energy level, i.e., E(0) = d, respectively. In Section 6, we consider the case of a > 1 and q = 2, give the finite time blowup of the solution to problem (1.1) at the arbitrarily positive initial energy level, i.e., E(0) > 0 and estimate the upper bound and lower bound of blowup time.
2 Preliminary results
2.1 Fractional Sobolev spaces
In this subsection, we first recall some necessary definitions and properties of the fractional Sobolev spaces, see also in [24], [33] for further detail.
In the following, for the sake of convenience, we denote by ‖ ⋅‖
p
the usual L
p
(Ω) norm for
Let W be the linear space of Lebesgue measurable functions u from
The space W is equipped with the norm
We shall work in the closed linear subspace
As shown in [34], an equivalent norm on W 0 is
and
Taking derivative of (2.2) with respect to t, we have
Specially, ⟨⋅, ⋅⟩ denotes the duality pairing between W 0 and its dual space W −s . Next, we state a functional property as following Proposition, proved in detail in [35].
Proposition 2.1.
For any
2.2 Potential well theory
In this subsection, we give some necessary definitions and lemmas related to potential well theory. We define the total, potential and Nehari energy functionals as follows
The stable set U and unstable set V are defined by
and
Finally, we define the depth of potential well
For the depth of the potential well d, we have the following knowledge.
Lemma 2.2.
(Depth of the potential well) The potential well depth is
where
Proof.
From the definition of d, for I(u) = 0, we have
which together with (2.10) gives
Then we have
Combining (2.8), (2.11) and I(u) = 0, there holds
which implies (2.9). □
Lemma 2.3.
(Non-increasing energy) Let u(x, t) be the weak solution to problem (1.1) with (u 0, u 1) ∈ W 0 × L 2(Ω). Then
Proof.
Testing the equation in (1.1) with u t , we obtain
Obviously, for any t ∈ [0, ∞), the energy is non-increasing. □
Lemma 2.4.
Let a > 0, (1.2), (1.3) and u ∈ W 0 \{0} hold, then:
For 0 < λ < ∞, there exists a unique λ* = λ*(u) such that
J(λu) is increasing on 0 < λ ≤ λ*, decreasing on λ* < λ < ∞ and takes the maximum at λ = λ*.
I(λu) > 0 for 0 < λ < λ*, I(λu) < 0 for λ* < λ < ∞, and I(λ*u) = 0.
Proof.
(i) From the definition of J(u), we know
Due to
and
(ii) Taking the derivative of (2.13) with respect to λ yields
We set
with (1.2), we know
and there exists
such that
i.e.,
(iii) By a direct calculation, (2.14) gives
(iv) The conclusion follows from
Hence, when 0 < λ < λ*, I(λu) > 0; when λ* < λ < ∞, I(λu) < 0; when λ = λ*, I(λ*u) = 0. □
Proposition 2.5.
(Gronwall inequality) Suppose y(t) ∈ L 1[0, T], if there exist constants b and c such that
Then
3 Local solution
We first give the definition of weak solution to problem (1.1).
Definition 3.1.
(Weak solution) A function u = u(x, t) is said to be a weak solution to problem (1.1), if u ∈ C([0, T], W 0) ∩ C 1([0, T], L 2(Ω)) ∩ C 2([0, T], W −s ) with u t ∈ L 2([0, T], W 0), and the initial data satisfy
such that
for any ϕ ∈ W 0 and a.e. t ∈ [0, T], where
For a given T > 0, consider the space
endowed with the norm
where the positive constant a was defined in (1.5). Next, we prove the existence and uniqueness of the local solution to problem (1.1). Firstly, we give an estimate of nonlinear term |u| p−2 u by Gâteaux derivative.
Lemma 3.2.
If (1.2) holds, for any u 1(x, t), u 2(x, t) with (x, t) ∈ Ω × [0, T], |u 1| + |u 2| > 0 and u 1 ≠ u 2, there holds
Proof.
Let S(u)≔|u| p−2 u. Define ς(x, t)≔u 1 − u 2 for (x, t) ∈ Ω × [0, T]. By the property of Gâteaux derivative, we know
for η 0 ∈ (0, 1), which together with the definition of Gâteaux derivative
gives
The proof is completed. □
In order to prove the local existence of the solution to the nonlinear problem (1.1) by the Banach fixed point theorem, we first convert problem (1.1) into the linearized problem. For the wave equation with nonlinear weak damping |u
t
|
q−2
u
t
, Georgiev and Todorova [36] inspired by Lions [37] gave a standard method for linearizing the nonlinear problem and proved the existence and uniqueness of the solution to the linearized problem. However, for the model equation in (1.1) considered in the present paper, the method proposed by [36], [37] does not work due to the appearance of the fractional Laplacian strong damping (−Δ)
s
u
t
and the fractional Kirchhoff term
Next, we give the following version of the linearized problem to (1.1) and consider the existence and uniqueness of its solution.
Lemma 3.3.
(Linearized problem to (1.1) with
Proof.
By [33], there exists a sequence
which solves the problem
for every η ∈ {ω j } and t ≥ 0. For j = 1, …, h, taking η = ω j in problem (3.6), we estimate the terms of the equation in (3.6). The first term becomes
For the second term, we get
For the third term, we obtain
Then by (3.7), (3.8) and (3.9), we obtain the following linear ordinary differential problem with unknown
where ψ
j
(t) = ∫
Ω|u|
p−2
uω
j
− |u
t
|
q−2
u
t
ω
j
dx ∈ C[0, T] due to
Taking
i.e.,
for every h ≥ 1. By Cauchy-Schwarz inequality and Hölder inequality, we estimate the first term on the right hand side in (3.11)
In particular, (3.5) implies that
for every t ∈ [0, T]. Due to (1.2), then
Therefore
further we get
Since
By Cauchy-Schwarz inequality and Hölder inequality, we estimate the last term on the right hand side in (3.11)
In order to deal with (3.16), we divide the range of the power q, i.e.,
Case I:
For the case
Meanwhile,
Case II:
For the case
where
and 0 < α < 1 is guaranteed by the condition
By Hölder inequality and Young inequality, we know (3.21) becomes
where
α is defined by (3.20), and the condition
Then for
where
then from Proposition 2.5, we obtain
Therefore, we know
Further, (3.26) becomes
By this uniform estimate and (3.6), we have
Therefore, up to a subsequence, we could pass to the limit in (3.6) satisfying the above regularity. Then the existence of weak solution to problem (3.4) can be obtained. Since v ∈ L ∞ ([0, T], W 0) and v t ∈ L 2([0, T], W 0), we get v ∈ C([0, T], W 0). Moreover since v t ∈ L 2([0, T], W 0)) and v tt ∈ L 2([0, T], W −s )), we have v t ∈ C([0, T], L 2(Ω)). And (3.4) implies that v tt ∈ C([0, T], W −s ). The following is the proof of the uniqueness. Arguing by contradiction, if v and w are two solutions of (3.4) sharing the same initial data. By subtracting the equations, we have
Testing (3.27) with v t − w t and integrating over [0, t] ⊂ [0, T], we have
By direct calculation of (3.28), there holds
By using (2.3), we know that (3.29) gives
From (3.30),
which together with (1.5) gives
Combining (3.31) and Gronwall inequality, we have
which yields v ≡ w. The proof is completed. □
Next, we give the theorem about the local existence of the solution to problem (1.1) based on Lemma 3.3.
Theorem 3.4.
(Local existence for
Then problem (1.1) admits a weak local solution over [0, T] satisfying
Proof.
We define the map Φ1, such that v = Φ1(u), where v is the unique solution to problem (3.4). Let (u 0, u 1) satisfy (3.1) and
then for any T > 0 we define
where
Next, we prove that
By the completeness of L 2(Ω) and W 0, there exists u ∈ L 2(Ω) such that u n → u strongly in L 2(Ω) as n → ∞, and u ∈ W 0 such that u n → u strongly in W 0 as n → ∞, i.e.,
That is
We denote
which together with (3.34) gives
By Hölder inequality and Young inequality, we know that (3.35) becomes
For the first term and the last term on the right hand side of (3.36), we argue it in the same spirit (although slightly differently) as for (3.15), (3.18) and (3.25) and get
for the case
for the case
for any time t ∈ (0, T], which means
i.e.,
Choosing T sufficiently small such that 1 − 2a −2(θ−1) R 4(θ−1) T > 0 and
then (3.41) gives
which shows that
i.e.,
Next, we prove that Φ1 is contractive. Now taking w
1 and w
2 in
Then, we have
Taking η = v t and integrating both sides of (3.43) on [0, t] ⊂ [0, T], we get
which gives
by Hölder inequality. Here, if w 1 = w 2 at some time t, we know
in (3.44). For the case w 1 ≠ w 2, by the property of Gâteaux derivative, we have
Then, by (3.45) and Young inequality, we know that (3.44) gives
Since w
1 and w
2 are taken from the space
In order to estimate the right hand side of (3.47), by the similar process of proving Lemma 3.2, we have
which gives
From the condition
By using Hölder inequality, we know that (3.48) becomes
where
Noticing the condition
Due to w
1 and w
2 being taken from the space
which means (3.52) gives
Here, from (3.51) and
Then, by interpolation inequality, we know that (3.53) gives
where
and we know 0 < ρ < 1 according to (3.54). Using Hölder inequality and Young inequality, we know that (3.55) becomes
where
And using Lemma 3.2, we obtain
for
By (3.58) and (3.59), we know (3.47) becomes
where ρ is defined by (3.56). Then, from (3.60), we have
i.e.,
For sufficiently small T, we know 1 − CT > 0 and
By the Banach fixed point theorem, there exists a unique local solution for problem (1.1) defined on [0, T]. Here, we can deduce that u
t
∈ L
q
([0, T], L
q
(Ω)). For the case
For the case
Hence, the proof of Theorem 3.4 is completed. □
4 Subcritical initial energy level E(0) < d
In this section, we prove the global existence, asymptotic behavior and blowup of the solution to problem (1.1) for subcritical initial energy level E(0) < d.
4.1 Global existence for subcritical initial energy E(0) < d
In this subsection, as an application of Galerkin method, we prove the global existence of the solution to problem (1.1) for subcritical initial energy level E(0) < d. First, we prove the invariance of set U for E(0) < d as follows.
Lemma 4.1.
(Invariant set U for E(0) < d). Let (1.2), (1.3) and (u 0, u 1) ∈ W 0 × L 2(Ω) hold. Then all the solutions to problem (1.1) with E(0) < d belong to U, provided u 0 ∈ U.
Proof.
Let u be a solution to problem (1.1) with E(0) < d, u 0 ∈ U and T be the maximum existence time of u(t). Arguing by contradiction, due to the continuity of I(u(t)) in t, we suppose that t* is the first time such that I(u(t*)) = 0 and I(u(t)) > 0 for t ∈ [0, t*). By Lemma 2.3, we have E(t) ≤ E(0) < d, which together with the definition of the depth of potential well d gives the following contradiction
Hence, the proof is completed. □
Theorem 4.2.
(Global existence for E(0) < d). Let (1.2), (1.3) and (u 0, u 1) ∈ W 0 × L 2(Ω) hold. Suppose that 0 < E(0) < d and u 0 ∈ U. Then problem (1.1) admits a global solution.
Proof.
Let ω j (x) be a sequence of eigenfunctions of the fractional Laplacian operator (−Δ) s , which is an orthogonal basis of L 2(Ω) and W 0. By Galerkin method, we construct approximate solution for problem (1.1)
satisfying
and
where h = 1, 2, …, m. Multiplying (4.1) by
Integrating (4.3) with respect to t, we have
Combining (2.5) with (2.6), we can obtain
From (2.8), we see that
By (4.2), we get E m (0) → E(0) and u m (0) → u 0 as m → ∞. From 0 < E(0) < d and u 0 ∈ U, we have u m (0) ∈ U and E m (0) < d, i.e.,
for sufficiently large m and t ≥ 0. Therefore, we get u m (t) ∈ U by the argument in the proof of Lemma 4.1. Since u m (t) ∈ U, i.e., I(u m (t)) > 0, then (4.4) gives
for sufficiently large m. Due to u m ∈ W 0, Proposition 2.1 gives
Hence, there exists a subsequence of {u m }, such that
for sufficiently large m, where
Hence, by (4.5), we can pass to the limit of (4.6) to obtain a weak solution to problem (1.1) satisfying (3.3). By (4.2), we obtain u(x, 0) = u 0 ∈ W 0 and u t (x, 0) = u 1 ∈ L 2(Ω). □
4.2 Asymptotic behavior for subcritical initial energy E(0) < d
In this subsection, we are concerned with the asymptotic behavior of the global solution to problem (1.1) with E(0) < d. First, we explore the relationship between the norm
Lemma 4.3.
Let (1.2), (1.3) and (u 0, u 1) ∈ W 0 × L 2(Ω) hold. Assume that u 0 ∈ U, E(0) < d, and u is the solution to problem (1.1). Then
for t ∈ [0, + ∞).
Proof.
By Lemma 2.3, we know that the energy E(t) is non-increasing. Since E(0) < d, combining (2.5), (2.6) and (2.8), we get
By Lemma 4.1 and u 0 ∈ U, we deduce u ∈ U, i.e., I(u) > 0 for t ∈ [0, ∞). Thus (4.8) shows that
for
Theorem 4.4.
(Asymptotic behavior for E(0) < d) Let (1.2), (1.3) and (u 0, u 1) ∈ W 0 × L 2(Ω) hold. If E(0) < d and u 0 ∈ U, then any global solution u to problem (1.1) has the following decay estimate
where κ, γ > 0 will be discussed later.
Proof.
By Theorem 4.2, we know that the solution u(t) of problem (1.1) exists globally.
Let
for ɛ > 0 and t ∈ [0, ∞). Next, we claim that for a suitable ɛ > 0, there exist two positive constants δ 1, δ 2 > 0 such that
Testing equation (1.1) with u, we obtain
Taking derivative of H(t) with respect to t, combining Lemma 2.3 with (4.10), we get
Next, we estimate each term to the right side of (4.11). Firstly, we deal with
where C 0 is defined in (2.10) and
By E(0) < d and Lemma 2.2, we have
Combining (2.5) and (4.12), for 0 < β 0 < 1, we can get
Substituting (4.13) into (4.11), we have
Next, we estimate the term ∫ Ω|u t | q−2 u t udx in (4.14). By Hölder inequality
where ρ, ϱ > 0 and
Let
Substituting (4.17) into (4.14), we have
Using Proposition 2.1, we have
Now for 0 < ξ 0 < 1 and (1.2), we have
such that
which means
and
Then combining (4.20) with (4.7), (4.19) becomes
For the term
which combining (4.22) indicates
where
and
Since 0 < ξ < 1, we choose large enough
Combining (4.21), (4.24) and (4.9), (4.23) becomes
Therefore, by Gronwall inequality, we can obtain
Further by (4.9), there holds
The proof is completed. □
4.3 Finite time blowup for subcritical energy E(0) < d
In this subsection, we prove the invariance of set V with E(0) < d, and then based on this property, we prove the finite time blowup of the solution.
Lemma 4.5.
(Invariant set V for E(0) < d). Let (1.2), (1.3) and (u 0, u 1) ∈ W 0 × L 2(Ω) hold. If E(0) < d and u 0 ∈ V, then all the weak solutions to problem (1.1) belong to V.
Proof.
The proof of is similar to Lemma 4.1, we omit it here. □
Lemma 4.6.
For I(u) < 0 and u ∈ W 0, there holds
and
satisfying (1.4).
Proof.
From u ∈ W 0 and I(u) < 0, (iv) of Lemma 2.4 can be applied to ensure that there exists a λ* such that 0 < λ* < 1 and I(λ*u) = 0. From the definition of d, we have
which means
The proof is completed. □
Next, we prove the finite time blowup of the solution to problem (1.1) for E(0) < d.
Theorem 4.7.
(Finite time blowup for E(0) < d with q = 2). Let (1.2), (1.3) and (u 0, u 1) ∈ W 0 × L 2(Ω) hold. Suppose that E(0) < d and u 0 ∈ V, then the solution to problem (1.1) blows up in finite time.
Proof.
Let u(x, t) be a weak solution to problem (1.1) with I(u 0) < 0 and E(0) < d. Arguing by contradiction, we suppose that u(x, t) is a global solution, i.e., T = +∞. For any T 0 > 0, we introduce the following auxiliary function
It is obviously that F(t) > 0 for all t ∈ [0, T 0]. From the continuity of F(t) with respect to t, it is easy to see that there exists ρ > 0 (independent of the choice of T 0) such that
Moreover, for t ∈ [0, T 0], direct calculation gives
and
Next, we estimate the right hand side of (4.27). By Cauchy-Schwarz inequality, we obtain
and
Combining (4.28) and Young inequality, we derive
Then (4.27) becomes
Set ϕ = u(⋅, t) in (3.2), using (2.7), we obtain
Then combining (4.26) and (4.30), we have
By (4.25), (4.29) and (4.31), we get
where
From (2.12) and q = 2, we get
hence
From (4.33) and (4.34), we obtain
for
then (4.35) gives
So (4.32) becomes
Let
for every t ∈ [0, T 0], which means that
for some finite time T > 0, i.e.,
The proof is completed. □
5 Critical initial energy level E(0) = d
In this section, we shall study the global existence, asymptotic behavior of global solution and blowup of solution to problem (1.1) at critical initial energy level E(0) = d.
5.1 Global existence for critical initial energy E(0) = d
Theorem 5.1.
(Global existence for E(0) = d). Let (1.2), (1.3) and (u 0, u 1) ∈ W 0 × L 2(Ω) hold. Suppose that E(0) = d and u 0 ∈ U, then problem (1.1) admits a global solution u with u ∈ L ∞ ([0, ∞), W 0) and u t ∈ L 2([0, ∞), W 0).
Proof.
We prove the theorem in two cases.
Case 1:
Let
for problem (1.1). From u
0 ∈ U, i.e., I(u
0) > 0 and (iv) in Lemma 2.4, we know λ* = λ*(u
0) > 1. It implies
Together with sufficiently large m, it is easy to see u m ∈ L ∞ ([0, ∞), W 0) and u mt ∈ L 2([0, ∞), W 0) from Theorem 4.2. The rest of the proof is similar to Theorem 4.2.
Case 2:
Let
for problem (1.1). By
which yields
Combining Theorem 4.2, we can deduce u m ∈ L ∞ ([0, ∞), W 0) and u mt ∈ L 2([0, ∞), W 0) for sufficiently large m. The rest proof is similar to Theorem 4.2. □
5.2 Asymptotic behavior for critical initial energy E(0) = d
Next, we give a lemma as follows to prove the asymptotic behavior of global solution to problem (1.1) with E(0) = d.
Lemma 5.2.
Let (1.2), (1.3) and (u 0, u 1) ∈ W 0 × L 2(Ω) hold. If E(0) = d, assume that u(x, t) is the solution to problem (1.1). Then we have
for t ∈ (0, ∞).
Proof.
Let u(t) be any nontrivial solution for problem (1.1) with E(0) = d. Arguing by contradiction, we suppose that
for t ∈ (0, ∞), which means
Theorem 5.3.
(Asymptotic behavior for E(0) = d) Let (1.2), (1.3) and (u 0, u 1) ∈ W 0 × L 2(Ω) hold. If E(0) = d and u 0 ∈ U, then any global (weak) solution u for problem (1.1) has the following decay estimate
where κ, γ > 0.
Proof.
We have proved that the solution u to problem (1.1) is global for E(0) = d in Theorem 5.1, then we claim that u ∈ U for t ∈ (0, ∞). Arguing by contradiction, we suppose that t 0 > 0 is the first time such that I(u(t 0)) = 0. By the definition of d, we get J(u(t 0)) ≥ d. By (2.12), we have
for any t
0 > 0. Hence we deduce J(u(t
0)) = d and
By the continuity of J(u) and I(u) with respect to t, we take a t 1 > 0 as the initial time, then u ∈ U for t > t 1. Therefore, by the same method with Theorem 4.4, we have E(t) ≤ κe−γt , t 0 ≤ t ≤ ∞. The proof is completed. □
5.3 Finite time blowup for critical initial energy E(0) = d
Theorem 5.4.
(Finite time blowup for E(0) = d with q = 2). Let (1.2), (1.3) and (u 0, u 1) ∈ W 0 × L 2(Ω) hold. Suppose that E(0) = d and u 0 ∈ V, then the solution to problem (1.1) blows up in finite time.
Proof.
Let u(t) be any local weak solution to problem (1.1) with E(0) = d > 0 and u 0 ∈ V. Next, we prove u(t) ∈ V for t ∈ (0, ∞). Arguing by contradiction, from the continuity of J(u) and I(u) with respect to t, there exists a sufficiently small t 0 > 0 such that I(u(t 0)) < 0 and E(t 0) > 0. Combining with (4.30), we have u t ≠ 0 for 0 < t ≤ t 0. Hence, by Lemma 2.3, there holds
By Lemma 4.5 and taking t = t 0 as the initial time, we get u ∈ V for t > t 0. The rest of the proof is similar to Theorem 4.7. □
6 Arbitrarily positive initial energy level E(0) > 0
6.1 Finite time blowup for arbitrarily positive initial energy E(0) > 0
In this part, we consider the case of q = 2. In order to prove the blowup of solution to problem (1.1) with E(0) > 0, we first give the following lemmas.
Lemma 6.1.
Let (1.2), (1.3) and (u 0, u 1) ∈ W 0 × L 2(Ω) and E(0) > 0 hold, and the initial data satisfy
hold, where C 0 > 0 is given in Proposition 2.1, then
is positive and strictly increasing provided u(t) ∈ V.
Proof.
Taking derivative of
By u(t) ∈ V, we know
Lemma 6.2.
(Invariant set V with E(0) > 0.) Let a > 1, (1.2), (1.3) and (u 0, u 1) ∈ W 0 × L 2(Ω) hold, and T be the maximum existence time of solution. If E(0) > 0, u 0 ∈ V and (6.1) hold, then all the weak solutions to problem (1.1) belong to V.
Proof.
We prove u(t) ∈ V by contradiction. Due to the continuity of I(u(t)) with respect to t, we suppose that there exists the first time t 0 ∈ (0, T) such that I(u(t 0)) = 0 and I(u(t)) < 0 for t ∈ [0, t 0). Hence, from Lemma 6.1, we know
for t ∈ [0, t 0) and C 0 > 0. Combining (2.5), (2.6), (2.8) and (2.12), we obtain
by a > 1 and (1.2). By Lemma 2.3 and the fact I(u(t 0)) = 0, (6.3) becomes
by a > 1 and (1.2). It is easy to see that (6.4) contradicts (6.1). The proof is completed. □
Theorem 6.3.
(Finite time blowup for E(0) > 0 with q = 2). Let a > 1, (1.2), (1.3) and (u 0, u 1) ∈ W 0 × L 2(Ω) hold. Suppose that E(0) > 0, u 0 ∈ V (6.1) hold, then the solution to problem (1.1) blows up in finite time.
Proof.
Arguing by contradition, we suppose that solution exists globally, i.e., T = +∞. Recalling auxiliary function F(t) as (4.25) and
where η 1 > 4 will be chosen later, and
By (6.3), we get
where (1.2) and a > 1 hold. Combining (6.5) and (6.6) and setting
we further derive
where
Hence
Let
and
which means that the function y(t) is concave. Therefore, there exists a time T 0 > 0 such that
i.e.,
which contradicts T max = +∞. Hence, the solution to problem (1.1) blows up in finite time. The proof is completed. □
6.2 Upper and lower bounds of blowup time for arbitrarily positive initial energy E(0) > 0
Finally, we estimate the upper bound and lower bound of blowup time of the blowup solution to problem (1.1) with E(0) > 0. In order to obtain the upper bound of blowup time, we first introduce the following lemma.
Lemma 6.4.
([39]) Support that a positive and twice-differentiable function Φ(t) satisfies the inequality
where t > 0 and α
1 > 0. If Φ(0) > 0 and Φ′(0) > 0, then there exists
Theorem 6.5.
(Upper bound of blowup time for E(0) > 0) Let a > 1, (1.2), (1.3) and (u 0, u 1) ∈ W 0 × L 2(Ω) hold. Assume that u 0 ∈ V, E(0) > 0, (u 0, u 1) > 0 and (6.1) hold. Let u be the solution to problem (1.1), which blows up at a finite time with
where F is defined in (4.25).
Proof.
Let u be the solution to problem (1.1). Assume the initial data satisfy (u 0, u 1) ∈ W 0 × L 2(Ω), u 0 ∈ V, E(0) > 0 and (6.1) hold. By Theorem 6.3, we know that the solution u blows up in finite time T. Recalling the auxiliary function F(t) given in (4.25), F(0) > 0 and F′(0) > 0 and the inequality as follow
where
By Lemma 6.4, we get the upper bound of blowup time T of solution to problem (1.1) with E(0) > 0, i.e.,
which yields (6.8). □
Theorem 6.6.
(Lower bound of blowup time for E(0) > 0) Assume that a > 1, (1.2), (1.3),
Then, one has
where C 0 is the constant given in (2.4), and
Proof.
By Lemma 2.3, we know E(t) ≤ E(0) and
Then (6.10) gives
and
By Hölder and Young inequalities, we know
As
Hence by (6.15), (6.14) becomes
Next, we substitute (6.12) and (6.13) into (6.16), separately.
which implies
Integrating (6.17) over [0, T], we have
By Theorem 6.3, there holds
which together with (6.18) gives
which implies
From (6.19), we obtain
which together with (6.20) yields (6.9). The proof is completed. □
Remark 6.7
(Open problems). In this section, we only consider the special case of non-degenerate Kirchhoff-type, i,e, a > 1 in (1.5), however the degenerate Kirchhoff-type case have still not been resolved. For the subcritical, critical and arbitrarily positive initial energy levels, we only established the finite time blowup of the solution for the case of linear weak damping, i.e., q = 2, and the case of nonlinear weak damping, i.e., q > 2 is still an open problem.
Funding source: NSERC (Canada) Grant
Award Identifier / Grant number: RGPIN-2019-05940
Funding source: Fundamental Research Funds for the Central Universities
Award Identifier / Grant number: 3072023GIP2401
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 12271122
Funding source: China Scholarship Council
Award Identifier / Grant number: 202306680038
Acknowledgements
The authors would like to appreciate the referees for taking their valuable time to provide constructive and valuable comments, which led to the improvement of our paper. Part of this work was done while Chao Yang was studying in the Faculty of Applied Mathematics, AGH University of Krakow, Poland.
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Research ethics: Not applicable.
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Informed consent: Not applicable.
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Use of Large Language Models, AI and Machine Learning Tools: None declared.
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Conflict of interest: The authors state no conflict of interest.
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Research funding: Shaohua Chen was supported by the NSERC (Canada) Grant RGPIN-2019-05940. Jiangbo Han was supported by the Fundamental Research Funds for the Central Universities (3072023GIP2401). Runzhang Xu was supported by the National Natural Science Foundation of China (12271122), and the Fundamental Research Funds for the Central Universities. Chao Yang was supported by the China Scholarship Council (No. 202306680038).
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Data availability: Not applicable.
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