Anomalous pseudo-parabolic Kirchhoff-type dynamical model

Xiaoqiang Dai; Jiangbo Han; Qiang Lin; Xueteng Tian

doi:10.1515/anona-2021-0207

Open Access Published by De Gruyter October 4, 2021

Anomalous pseudo-parabolic Kirchhoff-type dynamical model

Xiaoqiang Dai , Jiangbo Han , Qiang Lin and Xueteng Tian

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2021-0207

Abstract

In this paper, we study an anomalous pseudo-parabolic Kirchhoff-type dynamical model aiming to reveal the control problem of the initial data on the dynamical behavior of the solution in dynamic control system. Firstly, the local existence of solution is obtained by employing the Contraction Mapping Principle. Then, we get the global existence of solution, long time behavior of global solution and blowup solution for J(u₀) ⩽ d, respectively. In particular, the lower and upper bound estimates of the blowup time are given for J(u₀)<d. Finally, we discuss the blowup of solution in finite time and also estimate an upper bound of the blowup time for high initial energy.

Keywords: Pseudo-parabolic Kirchhoff-type equation; Global existence; Asymptotic behavior; Blowup

MSC 2010: 35B40; 35R11; 35K55

1 Introduction and main result

The paper is devoted to the study of an anomalous pseudo-parabolic Kirchhoff-type dynamical model as follows

(1.1) ut+M([u]s2)(−Δ)su+(−Δ)sut=|u|q−2u,in Ω×R+,u(x,0)=u0(x),in Ω,u(x,t)=0,in (RN∖Ω)×R0+,

where s ∈ (0, 1), N > 2s, Ω⊂ℝ^N is a bounded domain with Lipschitz boundary ∂Ω. The Kirchhoff function M(t)=t^λ−1 for t∈R0+ , here 1≤λ<NN−2s , and q satisfy 2λ<q≤2s∗ , where 2s∗ is the fractional critical exponent given by

2s∗:=2NN−2s.

And [u]_s is the Gagliardo seminorm of u defined by

[u]s:=∬R2N|u(x)−u(y)|2|x−y|N+2sdxdy12.

(−Δ)^s is the fractional Laplacian which, up to a normalization constant, is defined for any x ∈ ℝ^N

(−Δ)sφ(x):=2limε→0+∫RN∖Bε(x)φ(x)−φ(y)|x−y|N+2sdy,

for any φ∈C0∞(RN) , where B_ε(x) denotes the ball in ℝ^N with radius ε>0 centered at x ∈ ℝ^N. We can refer to [23, 24,35, 5, 36,34,22] for more details on nonlocal operators and nonlocal Sobolev spaces.

Problem (1.1) is a class of nonlocal fractional diffusion problem, which is related to the anomalous diffusion theory. A usual model for anomalous diffusion is the linear evolution equation involving the fractional Laplacian

∂tu+(−Δ)su=0,

which derives asymptotically from basic random walks models, see [2,21,38] and references therein. We denote by u(x, t) the probability of finding the particle at the point x at time t. Through a series of calculations, we can obtain ∂_tu(x, t)=−c_{n, s}(−Δ)^su(x, t) for a suitable c_{n, s} > 0, which shows that, for small time and space steps, the above probabilistic process approaches a fractional heat equation. Another nonlinear anomalous diffusion equation is the fractional porous medium equation ∂_tu+(−Δ)^s(u^m)=0 with 0 < s<1 and m > 0, which was first proposed by De Pablo et al. in [31]. Many important results on these equations have been obtained, see an overview in [40] and references therein.

To the best of our knowledge, fractional Laplacian operator and related equations have a growing wide utilization in many important fields, as explained by Caffarelli in [3] and Vázquez in [39]. In particular, the steady state of problem (1.1) without strong damping term, first proposed by Fiscella and Valdinoci in [12] by taking into account the nonlocal aspect of the tension arising from nonlocal measurements of the fractional length of the string, is a fractional version of the so-called stationary Kirchhoff model. Subsequently, the existence of weak solutions which solves the above stationary problem was obtained. Later, Fu and Pucci in [9] proved the existence of global solutions with exponential decay and showed the blow-up in finite time of solutions to the space-fractional diffusion equation

ut+(−Δ)su=|u|p−1u, in Ω×R+,

where Ω⊂ℝ^N is a bounded domain, p satisfies 1<p≤2s∗−1=(N+2s)/(N−2s) and N > 2s.

In recent years, much interest has grown on Kirchhoff-type problems, see for example [12, 27]. In these papers, to obtain the existence of weak solutions, the authors always assume that the Kirchhoff function M:R0+→R+ is a continuous and nondecreasing function and satisfies the following condition:

(1.2) there exists m0>0 such that M(t)≥m0 for all t∈R0+.

A typical example is M(t)=m₀+bt^m with m₀ > 0, b=0 for all t∈R0+ . Hence, we can divide the problem into degenerate and non-degenerate cases according to M (0)=0 and M (0)>0 respectively. For the non-degenerate Kirchhoff-type problem, we can refer to [10,29]. It is worthwhile pointing out that the degenerate case is rather interesting and is treated in well-known papers in Kirchhoff theory, see for example [4]. From a physical point of view, the fact that M (0)=0 means that the base tension of the string is zero. For some recent results in the degenerate case, see for instance [1,25,28,42,44]. In this regard, Pan et al. in [30] studied for the first time the degenerate Kirchhoff-type diffusion problem involving fractional p-Laplacian of following equation

(1.3) ut+[u]s,p(λ−1)p(−Δ)psu=|u|q−2u, in Ω×R+,

where Ω⊂ℝ^N, p < q<Np/(N−sp) with 1 < p<N/s and 1 ⩽ λ<N/(N−sp). They obtained the existence of a global solution by combining the Galerkin method with potential well theory to discover the control mechanics of the initial data on the dynamical behavior of the solution. Yang et al. in [50] studied the same problem with pλ<q<Np/(N−sp), they obtained the blow up of solutions by applying the concave method. Moreover, the authors estimated an upper bound of blow-up time in the sub-critical initial energy case J(u₀)<d and arbitrary positive initial energy case J(u₀)>0. Later, by implementing the same theory as shown in [30], for the initial boundary value problem of (1.3) with λ=1 and the more general nonlinearity f(u) instead of ∣∣u∣∣^q−2u, authors in [18] studied the existence and nonexistence of global weak solutions in the cases of J(u₀)<d, J(u₀)=d and J(u₀)>d, respectively. Moreover, the authors estimated an upper bound of blow-up time for low and high initial energies. Xiang et al. [43] considered the initial boundary value problem of the following Kirchhoff type equation

(1.4) ut+M[u]s2(−Δ)su=|u|p−2u, in Ω×R+,

where Ω⊂ℝ^N, M:[0, +∞) → [0, +∞) is a continuous function and there exist two constants m₀ and λ>1 such that M(σ) ⩾ m₀σ^λ−1, \;σ ∈ [0, +∞). For 1<λ<2N/(N − 2s), the local existence of nonnegative solutions of (1.4) is obtained by applying the Galerkin method. Moreover, the blowup conditions for nonnegative weak solution were obtained for J(u₀)<d. Then for the initial boundary value problem (1.4), Ding and Zhou [6] proved the global existence and finite time blowup of solution when J(u₀) ⩽ d for the case M(σ)=m₀σ^λ−1. Moreover, they showed that the nonnegative weak solution exists globally for λ ⩾ 2N/(N − 2s). Also Ding and Zhou in [7] gave the global existence and finite time blowup results with J(u₀)>d.

If s ↑ 1, M ≡ 1, then the equation in (1.1) reduces to the following equation

(1.5) ut−Δut−Δu=up, in Ω×R+.

In [48], Xu and Su used the family of the potential wells to prove the nonexistence of solutions with initial energy J(u₀) ⩽ d, and obtained finite time blowup with high initial energy J(u₀)>d by comparison principle. Later, Xu et al. in [49] discussed the same problem, they established a new finite time blowup theorem for problem (1.5) and estimated the upper bound of blowup time for J(u₀)>0. Previously, Liu and Zhao in [21] considered the initial-boundary value problem u_t−Δu=f(u) with initial data J(u₀)<d for I(u₀)<0 and I(u₀)⩾0, and initial data J(u₀)=d for I(u₀)⩾0. Xu in [46] studied the same problem with critical initial data J(u₀)=d, I(u₀)<0. A powerful technique for treating the above problem is the so-called potential well method, which was established by Payne and Sattinger in [27]. Since then, the potential well method has been widely used to study the well-posedness of solution for evolution equations, such as [18,46,47]. Gazzola and Weth in [13] studied the initial-boundary value problem of u_t−Δu=∣∣u∣∣^p−1u, they proved finite time blow-up of solutions with high initial energy J(u₀)>d by the comparison principle and variational methods. Recently, the threshold results of global existence and finite time blowup for several types of pseudo-parabolic equations were established in [33,42].

It is worthy pointing out that the Kirchhoff-type parabolic problem

(1.6) ut−M∫RN|∇u|2dxΔu=|u|q−1u, in Ω×R+

was studied by Han et al. [14] where the global existence and finite time blowup of solutions were proved in the sub-critical, critical and super-critical cases. Here Ω⊂ℝ^N is a bounded domain, M(τ)=a+bτ with a, b > 0. In [15], by some differential inequalities, the authors investigated the upper and lower bounds of blowup time for the weak solution to (1.6).

Motivated by the above works, the main objective of this paper is to consider a more complicated case of the problem (1.4) studied in [6,7,44] by taking damping term (−Δ)^su_t in the fractional setting. More precisely, we focus on the local and global well-posedness of degenerate Kirchhoff’s model of parabolic type (1.1), by using potential well theory and concave function method.

The outline of this paper is as follows. In Section 2, we recall some necessary definitions and properties of the fractional Sobolev spaces and introduce the family of potential wells. In Section 3, we prove the existence of local solutions for problem (1.1). In Section 4, we prove the global existence, the finite time blow-up, the asymptotic behavior for problem (1.1) and give the lifespan estimates of the blowup solution with J(u₀)<d. In Section 5, we parallelly extend some conclusions for the sub-critical initial energy case to the critical initial energy. In Section 6, by constructing a new unstable set and using some differential inequality techniques, we also study the finite time blowup solution for problem (1.1) and give the upper bound estimate of the blowup time at arbitrary positive initial energy level.

2 Preliminaries

2.1 Functional spaces

In this section, we first recall some necessary definitions and properties of the fractional Sobolev spaces, see also [5, 11] for further details.

Throughout the paper, s ∈ (0, 1), N > 2s and 2λ<q≤2s∗ . We denote Q=R2N∖G , where G=C(Ω)×C(Ω)⊂R2N , and C(Ω)=RN∖Ω . W is a linear space of Lebesgue measurable functions from ℝ^N to ℝ such that the restriction to Ω of any function u in W belongs to L²(Ω) and

∬Q|u(x)−u(y)|2|x−y|N+2sdxdy<∞.

The space W is equipped with the norm

∥u∥W:=∥u∥L2(Ω)2+∬Q|u(x)−u(y)|2|x−y|N+2sdxdy12,

It is easy to get that ∣∣·∣∣_W is a norm on W. We shall work in the closed linear subspace

(2.1) W0:=u∈W:u(x)=0 a.e. in RN∖Ω.

By [35], we can get an equivalent norm on W₀ defined as

∥v∥W0:=∬Q|v(x)−v(y)|2|x−y|N+2sdxdy12.

We put

(u,v)X0:=(u,v)+(u,v)W0,∥u∥X02:=∥u∥22+∥u∥W02,

where ∥⋅∥X0 is an equivalent norm over W₀.

Lemma 2.1

([35, lemma 6]) Let K:ℝ^N∖{0} → (0, ∞) satisfy K(x)=∣∣x∣∣^−(N+2s). Then there exists a positive constant C₀=C₀(N, 2, s) such that for any v ∈ W₀ and v∈[1,2s∗] ,

∥v∥Lv(Ω)2≤C0∬Ω×Ω|v(x)−v(y)|2|x−y|N+2sdxdy≤C˜∬Q|v(x)−v(y)|2|x−y|N+2sdxdy.

Lemma 2.2

(Gronwall inequality). Assume y(t) ∈ L¹[0, T], and there exist constants a and b such that

(2.2) y(t)≤a+b∫0Ty(z)dz,0≤t≤T,

(2.3) y(t)≤aebt,0≤t≤T,

Definition 2.1

(Weak solution). A function u ∈ L^∞(0, ∞; W₀) is said to be a weak solution of problem (1.1), if u_t ∈ L²(0, ∞; X₀) and u₀ ∈ W₀ for a.e. t > 0,

(2.4) ∫Ω∂tuvdx+〈u,v〉W0+ut,vW0=∫Ω|u|q−2uϕdx,

where

(ut,ϕ)W0:=∬Qut(x)−ut(y)⋅v(x)−v(y)|x−y|N+2sdxdy

and

〈u,v〉W0:=M(∥u∥W02)∬Qu(x)−u(y)⋅v(x)−v(y)|x−y|N+2sdxdy

for any v ∈ W₀.

Then we define the potential energy functional of problem (1.1) as follows

(2.5) J(u):=12λ∥u∥W02λ−1q∥u∥qq,

the Nehari functional

(2.6) I(u):=∥u∥W02λ−∥u∥qq.

Moreover, there holds

(2.7) ∫0tuτX02dτ+J(u)≤Ju0.

Next we introduce the Nehari manifold

N:={u∈X0∣∣I(u)=0,∥u∥X0≠0}.

Furthermore, we set

N + := { u ∈ X 0 ∣∣ I ( u ) > 0 } , N − := { u ∈ X 0 ∣∣ I ( u ) < 0 } .

The potential well depth is defined as

(2.8) d:=infu∈NJ(u).

Further we give some sets as follows

Wp:={u∈X0∣∣J(u)<d,I(u)>0}∪{0},Vp:={u∈X0∣∣J(u)<d,I(u)<0}.

2.2 Family of potential wells

In this section, we shall introduce a family of potential wells W_δ and V_δ, and give a series of their properties for problem (1.1). Firstly, let the definitions of functionals J(u), I(u) and the potential well W_p with its depth d given above hold. Next, we give some properties of above sets and functionals as follows.

Lemma. 2.3

Let u ∈ X₀ and ∥u∥X0≠0 . Then

lim_θ → 0J(θ u)=0, lim_{θ → +∞}J(θ u)=−∞.
On the interval 0<θ<∞, there exists a unique θ*=θ*(u), such that ddθJ(θu)|θ=θ∗=0 .
J(θ u) is increasing for 0 ⩽ θ ⩽ θ*, decreasing for θ* ⩽ θ<∞ and take the maximum at θ=θ*.
I(θ u) > 0 for 0<θ<θ*, I(θ u) < 0 for θ*<θ<∞ and I(θ*u)=0.

Proof.

By (2.5), we know

J(θu)=θ2λ2λ∥u∥W02λ−θqq∥u∥qq,

which gives

limθ→0J(θu)=0

and

limθ→∞J(θu)=−∞

by the fact that q > 2λ⩾2.
An easy calculation shows that

(2.9) ddθJ(θu)=θ2λ−1∥u∥W02λ−θq−1∥u∥qq,

let ddθJ(θu)=0 , we get

θ∗=∥u∥W02λ∥u∥qq1q−2λ,

which leads to the conclusion.
By a direct calculation (2.9) gives ddθJ(θu)>0 for 0<θ<θ*, ddθJ(θu)<0 for θ*<θ<∞. Hence, the conclusion of (iii) holds.
Since

I(θu)=θ2λ∥u∥W02λ−θq∥u∥qq=θddθJ(θu),

then the conclusion follows immediately.

□

Now, for δ>0,we define

I δ ( u ) := δ ∥ u ∥ W 0 2 λ − ∥ u ∥ q q , N δ := { u ∈ X 0 | I δ ( u ) = 0 , ∥ u ∥ X 0 ≠ 0 }

and

(2.10) d(δ):=infu∈NδJ(u).

Further we set

Wδ:={u∈X0|Iδ(u)>0,J(u)<d(δ)}∪{0},Vδ:={u∈X0|Iδ(u)<0,J(u)<d(δ)}

and

r(δ):=δC∗q1q−2λ,

where C_* is the embedding constant for W₀↪ L^q(Ω).

Lemma 2.4

Let u ∈ X₀ and 0<δ<q2λ , then we have

I_δ(u)≥0, provided 0<∥u∥W0≤r(δ) . Particularly, I(u)≥0 when 0<∥u∥W0≤r(1) .
∥u∥W0>r(δ) , provided I_δ(u) < 0. Particularly, ∥u∥W0>r(1) when I(u) < 0.
∥u∥W0≥r(δ) or ∥u∥W0=0 , provided I_δ(u)=0. Particularly, ∥u∥W0≥r(1) or ∥u∥W0=0 when I(u)=0.
J(u) > 0 for 0<δ<q2λ , provided I_δ(u)=0 and ∥u∥W0≠0 .

Proof.

From 0<∥u∥W0≤r(δ) , it follows that

∥u∥qq≤C∗q∥u∥W0q=C∗q∥u∥W02λ∥u∥W0q−2λ≤δ∥u∥W02λ,

that is I_δ(u)=0.
It is easy to see ∥u∥W0≠0 by I_δ(u) < 0. Thus from

δ∥u∥W02λ<∥u∥qq≤C∗q∥u∥W0q=C∗q∥u∥W02λ∥u∥W0q−2λ,

it implies ∥u∥X0>r(δ) .
When ∥u∥W0=0 , we can obviously get I_δ(u)=0. When ∥u∥W0≠0 and I_δ(u)=0, from

δ∥u∥W02λ=∥u∥qq≤C∗q∥u∥W02λ∥u∥W0q−2λ,

it follows that ∥u∥W0≥r(δ) .
By combining Lemma 2.4 (iii) and I_δ(u)=0, there holds

J(u)=12λ−δq∥u∥W02λ+δq∥u∥W02λ−1q∥u∥qq=12λ−δq∥u∥W02λ+1qIδ(u)=12λ−δq∥u∥W02λ.

Obviously, (iv) follows.

□

Lemma 2.5

Let δ>0, then the properties of d(δ) can be summarized as follows:

d(δ)=a(δ)r^2λ(δ) for 0<δ<q2λ and a(δ)=12λ−δq .
limδ→0d(δ)=0,limδ→q2λd(δ)=0 and d(δ)<0 for δ>q2λ .
d(δ) is increasing for 0<δ⩽1, decreasing for 1≤δ≤q2λ and takes the maximum at δ=1.

Proof.

When u∈Nδ , then from lemma 2.4 (ii) it gives ∥u∥X0≥r(δ) . Further by (2.8) and

J(u)=12λ−δq∥u∥W02λ+1qIδ(u)=12λ−δq∥u∥W02λ+1qIδ(u)=a(δ)∥u∥W02λ≥a(δ)r2λ(δ),

it follows that d(δ)=a(δ)r^2λ(δ).
By the conclusion of (i), obviously, (ii) holds.
For any u∈Nδ′′ , we shall prove that for all 1<δ″<δ′<q/(2λ) or 0<δ′<δ″<1, there exist ε(δ′, δ″)>0 and v∈Nδ′ satisfying J(u) − J(v)>ε(δ′, δ″). Actually, we can define θ(δ) by u∈Nδ′′ , then I_δ(θ(δ)u)=0 and θ(δ″)=1. Taking z(θ)=J(θ u), we get

ddθz(θ)=1θ(1−δ)∥θu∥W02λ+Iδ(θu)=θ2λ−1(1−δ)∥u∥W02λ.

Let v=θ(δ′)u, then v∈Nδ′ .

For 0<δ′<δ″<1, we have

J(u)−J(v)=z(1)−zθδ′=∫θδ′1ddθ(z(θ))dθ=∫θδ′1θ2λ−1∥u∥W02λ−θq−1∥u∥qqdθ=∫θδ′11θ∥θu∥W02λ−∥θu∥qqdθ=∫θδ′11θ(1−δ)∥θu∥W02λ+δ∥θu∥W02λ−∥θu∥qqdθ=∫θδ′11θ(1−δ)∥θu∥W02λdθ=∫θδ′1(1−δ)θ2λ−1∥u∥W02λdθ>∫θδ′11−δ′′r2λδ′′θ2λ−1δ′′dθ=1−δ′′r2λδ′′θ2λ−1δ′1−θδ′≡εδ′′,δ′′>0

For 1<δ′′<δ′<q2λ , we have

J(u)−J(v)=z(1)−zθδ′=∫θδ′1ddθ(z(θ))dθ=∫θδ′1θ2λ−1∥u∥W02λ−θq−1∥u∥qqdθ=∫θδ′11θ∥θu∥W02λ−∥θu∥qqdθ=∫θδ′11θ(1−δ)∥θu∥W02λ+δ∥θu∥W02λ−∥θu∥qqdθ=∫θδ′11θ(1−δ)∥θu∥W02λdθ=∫1θδ′(δ−1)θ2λ−1∥u∥W02λdθ>∫1θδ′δ′′−1r2λδ′′θ2λ−1δ′′dθ=δ′′−1r2λδ′′θ2λ−1δ′′θδ′−1≡εδ′,δ′′>0.

Therefore, the conclusion of (iii) is proved. □

Lemma 2.6

For u ∈ X₀ and 0<δ1<1<δ2<q2λ , if 0 < J(u) < d and d(δ₁)=d(δ₂)=J(u). Then I_δ(u) keeps the invariance of sign for δ ∈ (δ₁, δ₂).

Proof. J(u) > 0 implies ∥u∥W0≠0 . If the sign of I_δ(u) is changeable for δ₁<δ<δ₂, by the continuity of I_δ(u) in δ, then there exists a δ‾∈(δ1,δ2) such that Iδ‾(u)=0 . Therefore, by (iii) of Lemma 2.5 and (2.10), it follows that J(u)≥d(δ‾) , which obviously contradicts d(δ1)=d(δ2)=J(u)<d(δ‾) . □

3 Existence and uniqueness of local solution

Inspired by [38], in which Taniguchi considered the existence of a local solution to a Kirchhoff-type wave equation with damping. In this section, we shall prove the local well-posedness of solution to the Kirchhoff-type pseudo-parabolic equation of the form (1.1).

For a given T > 0, we consider the space H=C([0,T],X0) endowed with the norm

∥u∥H:=maxt∈[0,T]∥u∥X0.

In the following, the existence and uniqueness of solution for the linear problem corresponding to (1.1) is proved.

Lemma 3.1

For every T > 0, every u∈H and u₀ ∈ X₀, there exists a unique v satisfying

(3.1) v∈C[0,T],X0∩C1[0,T],L2(Ω),vt∈L2[0,T],X0,

which solves the linear problem

(3.2) vt+M(∥u∥W02)(−Δ)sv+(−Δ)svt=|u|q−2u,(x,t)∈Ω×R+v(x,0)=u0,x∈Ω,v(x,t)=0,(x,t)∈(RN∖Ω)×R0+,

Proof. The assertion follows from an application of the Galerkin method. By [36], for every h=1 let W_h=Span{ω₁, ···, ω_h}, where {ω_j} is the orthogonal complete system of eigenfunctions of (−Δ)^s in W₀ such that ∥ωj∥W0=1 and ∣∣ω_j∣∣₂=1 for all j. Then, {ω_j} is orthogonal and complete in L²(Ω) and W₀; denote by {λ_j} the related eigenvalues repeated according to their multiplicity. Let

u 0 h = ∑ j = 1 h ∫ Ω u 0 ω j ω j ,

so that u0h∈Wh,u0h→u0 in W₀ as h → ∞. For all h=1 we seek h functions γ1h,⋯,γhh∈C1[0,T] such that

(3.3) vh(t)=∑j=1hγjh(t)ωj,

solving the problem

(3.4) ∫Ωv˙h(t)+M(∥u∥W02)(−Δ)svh+(−Δ)sv˙h−|u|q−2uηdx=0,vh(0)=u0h,

for every η ∈ W_h and t=0. For j=1, ···, h, taking η=ω_j in (3.4) yields the following Kirchhoff fractional Laplacian problem for a linear ordinary differential equation with unknown γjh

(3.5) γ˙jh(t)+M([u]s2)λjγjh(t)+λjγ˙jh(t)=ψj(t),γjh(0)=∫Ωu0ωj,

where ψ_j(t)=∫_Ω∣∣u∣∣^q−2uω_jdx ∈ C[0, T]. For all j, the above Kirchhoff fractional Laplacian problem yields a unique global solution γjh∈C1[0,T] . In turn, this gives a unique v_h defined by (3.3) and satisfying (5.5). In particular, (3.3) implies that v˙h(t)∈W0 for every t ∈ [0, T], so that fractional Sobolev inequality entails

(3.6) V ˙ h ( t ) q ≤ c 1 V ˙ h ( t ) W 0 .

Next, the proof is divided into the following two cases.

Case 1: M(∥u∥W02)≥m0>0 for any u ∈ W₀, where m₀ is a constant.

Taking η=v˙h(t) into (5.5) and integrating over [0, t]⊂[0, T], we obtain

(3.7) 12M∥u∥W02vh(t)W02+∫02v˙h(t)X02dτ=∫0t∫Ω|u|q−2uv˙h(τ)dxdτ+12Mu0W02u0hW02+∫0tu,utW0M′∥u∥W02vh(t)W02dτ,

for every h=1. Since u∈H , ∥u∥X0 is bounded. We estimate the last term in the right-hand side thanks to Hölder and Young inequalities

(3.8) ∫0t∫Ω|u|q−2uv˙h(τ)dxdτ≤∫0t∥u∥2N(q−1)N+2sq−1v˙h(τ)2NN−2sdτ≤∫0tc∥u∥W0q−1v˙hW0dτ≤cT+∫0t12v˙h(τ)X02dτ,

where c > 0 and represent different constants between different lines.

By combining (3.7), (3.8) and Hölder inequality, we obtain

12M(∥u∥W02)∥vh(t)∥W02+∫0t12∥v˙h(τ)∥X02dτ≤cT+M(∥u0∥W02)∥u0h∥W02+∫0t(u,ut)W0M′(∥u∥W02)∥vh∥W02dτ≤cT+M(∥u0∥W02)∥u0h∥W02+∫0t∥u∥W0∥ut∥W0M′(∥u∥W02)∥vh∥W02dτ≤L+∫0t∥u∥W0∥ut∥W0M′(∥u∥W02)∥vh∥W02dτ,

for every h=1, where L:=cT+M(∥u0∥W02)∥u0h∥W02 is a constant. Further by u∈H , we can deduce that there exists a L₁ such that ∥u∥W0∥ut∥W0M′(∥u∥W02)≤L1 . Hence from M(∥u∥W02)≥m0>0 and (3.9) we derive

m 0 2 ∥ v h ( t ) ∥ W 0 2 ≤ L + L 1 ∫ 0 t ∥ v h ∥ W 0 2 d τ .

By Gronwall’s inequality, we obtain

(3.9) ∫ 0 t ∥ v h ∥ W 0 2 d τ ≤ L L 1 ( e 2 L 1 m 0 t − 1 ) ,

then (3.9) yields that

(3.10) 12M∥u∥W02vh(t)W02+∫0t12v˙h(τ)X02dτ≤L+Le2L1m0T−1:=CT,

where C_T > 0 is independent of h. By this uniform estimate, the embedding W₀↪ L²(Ω) and using (3.4), we have

{vh}is bounded inL∞([0,T],W0);

{v˙h}is bounded inL2([0,T],X0).

Case 2: There is at least a ũ ∈ W₀ such that M (∣∣ũ∣∣_W₀²)=0.

Taking η=v_h(t) into (3.4) and integrating over [0, t]⊂[0, T], we obtain

(3.11) 2∫0tM∥u˜∥W02vhW02dτ+vhX02=u0hX02+2∫0t|u˜|q−2u˜vhdτ.

Since u˜∈H,∥u˜∥X0 is bounded. We estimate the last term in the right-hand side thanks to Hölder and Young inequalities

(3.12) 2∫0t∫Ω|u˜|q−2u˜vh(τ)dxdτ≤∫0tc∥u˜∥W0q−1vhW0dτ≤cT+∫0tvh(τ)X02dτ.

Substituting (3.12) into (3.11), we have

(3.13) 2∫0tM∥u˜∥W02vhW02+vhX02≤u0hX02+cT+∫0tvh(τ)X02dτ=A1+∫0tvh(τ)X02dτ,

where A1:=∥u0h∥X02+cT is a constant.

By using Gronwall’s inequality again, we have

∫ 0 t ∥ v h ( τ ) ∥ X 0 2 d τ ≤ A 1 ( e t − 1 ) ,

then

(3.14) vhX02≤A1+A1eT−1:=AT.

{vh} is bounded in L∞([0,T],X0).

Therefore, up to a subsequence, we may pass to the limit in (3.4) and obtain a weak solution v of (3.4) with the above regularity. Then by v ∈ L^∞([0, T], X₀) we deduce that v ∈ L²([0, T], X₀), which together with v˙∈L2([0,T],X0) gives that v∈W1,2([0,T],X0) , here W1,2 denotes the Sobolev space consisting all functions v ∈ L²([0, T], X₀) such that v˙∈L2([0,T],X0) . Further by Theorem 2 in [8, Chapter 5]evans we can derive that v ∈ C ([0, T], X₀). Naturally, it follows that v ∈ C ([0, T], X₀) and v ∈ C ([0, T], L²(Ω)). Finally, from (3.2) we have v˙∈C1([0,T],L2(Ω)) . The existence of v solving (3.2) and satisfying (3.1) is so proved.

Uniqueness follows arguing for contradiction: if v and ω are two solutions of (3.2) which share the same initial data, by subtracting the equations and testing with v_t−ω_t, instead of (3.7) we can get

12M([u]s2)∥v(t)−ω(t)∥W02+∫0t∥vt(τ)−ωt(τ)∥X02dτ=0,

which immediately yields ω ≡ v. The proof of the lemma is now complete. □

Next, we establish local existence and uniqueness of (1.1).

Theorem 3.1

Let u₀ ∈ X₀, then there exist T > 0 and a unique local solution of (1.1) over [0,T].

Proof. Let R2:=2m0(M(∥u0∥W02)∥u0∥W02) and for any T > 0 we consider

MT:={u∈H:u(0)=u0 and ∥u∥H2≤R2}.

By lemma 3.1, for any u∈MT , we may define v≔ Φ(u), being v the unique solution to problem (3.2). We claim that, for a suitable T > 0 and given u∈MT , Φ is a contractive map satisfying Φ(MT)⊆MT . Moreover, the corresponding solution v=Φ(u) satisfies for all t ∈ (0, T] the energy identity

(3.15) 12M∥u∥W02∥v∥W02+∫0tvt(τ)X02dτ=12Mu0W02u0W02+∫0tu,utW0M′∥u∥W02∥v∥W02dτ+∫0t∫Ω|u(τ)|q−2u(τ)vt(τ)dxdτ.

Next, we still divide the proof to two cases corresponding to Lemma 3.1.

Case 1: M(∥u∥W02)≥m0>0 for any u ∈ W₀, where m₀ is a constant.

For the last term on the right-hand side of (3.15), we argue in the same spirit (although slightly differently) as for (3.8) and we get

(3.16) ∫0t∫Ω|u(τ)|q−2u(τ)vt(τ)dxdτ≤c∫0t∥u(τ)∥2n(q−1)n+2sq−1vt(τ)2nn−2sdτ≤c∫0t∥u(τ)∥W0q−1vt(τ)W0dτ≤cTR2(q−1)+∫0tvtX02dτ

for all t ∈ (0, T]. Combining (3.15) with (3.16) we can get

(3.17) 12M∥u∥W02∥v∥W02≤cTR2(q−1)+12Mu0W02u0W02+∫0tu,utW0M′∥u∥W02∥v∥W02dτ≤L2+L1∫0t∥v∥W02dτ,

where L2:=cTR2(q−1)+12M(∥u0∥W02)∥u0∥W02 is a constant. By using Gronwall’s inequality, it gives

∥v∥W02≤2L2m0e2L1m0t,

∫0t∥v∥W02dτ≤L22L1(eL1m0t−1).

Then taking the maximum over [0, T] gives

(3.18) m0∥v∥W02≤M∥u∥W02∥v∥W02≤cTR2(q−1)+Mu0W02u0W02+L2eL1m0T−1.

Choosing T sufficiently small, we get ∥v∥H2≤R2 .

Case 2: There is at least a ũ ∈ X₀ such that M (∣∣ũ∣∣_W₀²)=0.

In this regard, let R2=∥u0∥X02 and for any T > 0 consider

MT={u∈H:u(0)=u0 and ∥u∥H2≤R2}.

Similar to (3.11) in lemma 3.1, the corresponding solution v=Φ(u) satisfies for all t ∈ (0, T] the energy identity

(3.19) 2∫0tM∥u˜∥W02∥v∥W02dτ+∥v∥X02=u0X02+2∫0t|u˜|q−2u˜v(τ)dτ,

(3.20) 2∫0t∫Ω|u˜|q−2u˜v(τ)dxdτ≤cTR2(q−1)+∫0t∥v(τ)∥X02dτ,

then

(3.21) ∫0tM∥u˜∥W02∥v∥W02+∥v∥X02≤u0X02+cTR2(q−1)+∫0t∥v(τ)∥X02dτ=A2+∫0t∥v(τ)∥X02dτ,

where A2:=∥u0∥X02+cTR2(q−1) is a constant. By Gronwall’s inequality

∫0t∥v(τ)∥X02dτ≤A2(et−1).

(3.22) ∥v∥X02≤u0X02+cTR2(q−1)+A2eT−1.

Choosing T sufficiently small, we get ∥v∥H2≤R2 .

Combining Case 1 and Case 2, we show that Φ(MT)⊆MT . Next we prove Φ is a contraction. Now take ω₁ and ω₂ in MT , subtracting the two equations (3.2) for v₁=Φ(ω₁) and v₂=Φ(ω₂), and setting v=v₁−v₂ we obtain for all η ∈ W₀ and a.e. t ∈ [0, T]

(3.23) v t , η + M [ u ] s 2 ( − Δ ) s v , η + ( − Δ ) s v t , η = ∫ Ω ω 1 ( t ) q − 2 ω 1 ( t ) − ω 2 ( t ) q − 2 ω 2 ( t ) η = ∫ Ω ζ ( t ) ω 1 ( t ) − ω 2 ( t ) η ,

where ς=ς(x, t)=0 is given by Lagrange Theorem so that ς(t) ⩽ (q−1)(∣∣ω₁(t)∣∣+∣∣ω₂(t)∣∣)^q−2. Therefore, by taking η=v_t in (3.23) and arguing as above, we obtain

∥Φ(ω1)−Φ(ω2)∥MT2=∥v∥MT2≤ζ∥ω1−ω2∥MT2

for some 0<ζ<1 provided T is sufficiently small. This proves the claim. By the Contraction Mapping Principle, there exists a unique weak solution to problem (1.1) defined on [0,T]. The main statement of Theorem 3.1 is thus proved. □

4 Sub-critical initial energy J(u₀)<d

In this section, we prove the invariance of some sets under the flow of problem (1.1).

Definition 4.1

(Maximal existence time). Let u(t) be a weak solution of problem (1.1). We define the maximal existence time T_max of u(t) as follows:

If u(t) exists for 0 ⩽ t<∞, then T_max=∞.
If there exists a t₀ ∈ (0, ∞) such that u(t) exists for 0 ⩽ t<t₀, but doesn't exists at t=t₀, then T_max=t₀.

Lemma 4.1

(Invariant sets when J(u₀)<d). Assume that u₀ ∈ X₀, 0 < e<d, δ₁<δ₂ are the two roots of equation d(δ)=e for 0<δ1<1<δ2<q2λ , T_max is the maximal existence time of u(t). Then

All weak solutions u of problem (1.1) with J(u₀)=e belong to W_δ for δ₁<δ<δ₂, 0 ⩽ t<T_max, provided I(u₀)>0.
All weak solutions u of problem (1.1) with J(u₀)=e belong to V_δ for δ₁<δ<δ₂, 0 ⩽ t<T_max, provided I(u₀)<0.

Proof.

(i) Let u(t) be any weak solution of problem (1.1) with J(u₀)=e, I(u₀)>0 or ∥u0∥X0=0 . T_max is the existence time of u(t). If ∥u0∥X0=0 , then u₀(x) ∈ W_δ. If I(u₀)>0 then from Lemma 2.6, it follows I_δ(u₀)>0 and J(u₀)<d(δ). Then u₀(x) ∈ W_δ for δ₁<δ<δ₂ and 0 < t<T_max. Arguing by contradiction, by the continuity of I(u (t)) in t, we suppose that there exists a first time t₁ ∈ (0, T_max) and δ₀ ∈ (δ₁, δ₂) such that u(t1)∈∂Wδ0 , i.e., Iδ0(u(t1))=0,∥u∥X0≠0 or J(u (t₁))=d(δ₀). From

(4.1) ∫0t∥u(τ)∥X02dτ+J(u)≤Ju0<d(δ),δ1<δ<δ2,0≤t<Tmax,

we can see that J(u (t₀))≠ d(δ₀). If Iδ0(u(t0))=0,∥u∥X0≠0 , then by the definition of d(δ) we have J(u (t₀)) ⩾ d(δ₀), which contradicts (4.1).

(ii) Let u(t) be any weak solution of problem (1.1) with J(u₀)=e, I(u₀)<0. From J(u₀)=e, I(u₀)<0 and Lemma 2.6, it follows I_δ(u₀)<0 and J(u₀)<d(δ). Then u₀(x) ∈ V_δ for δ₁<δ<δ₂. We prove u(t) ∈ V_δ for δ₁<δ<δ₂ and 0 < t<T_max. Arguing by contradiction, by the continuity of I(u (t)) with respect to t, we suppose that there exists a first time t₁ ∈ (0, T_max) and δ₀ ∈ (δ₁, δ₂) satisfying u(t1)∈∂Vδ0 , i.e., Iδ0(u(t1))=0 or J(u (t₁))=d(δ₀). However, from (4.1) we can deduce that J(u (t₁))≠ d(δ₀). Besides for Iδ0(u(t1))=0 , we have Iδ0(u(t))<0 for 0 ⩽ t<t₁. Then by (ii) of Lemma 2.4, it follows that ∥u(t1)∥X0>r(δ0) . Obviously, J(u (t₁))≠ d(δ₀) and ∥u(t1)∥X0≥r(δ0) , this contradicts (4.1). □

Remark 4.1

If in Lemma 4.2 the assumption J(u₀)=e is replaced by 0 < J(u₀) ⩽ e, then the conclusion of Lemma 4.1 also holds.

4.1 Global existence and finite time blowup of solution

In this section, we prove a threshold result of global existence and nonexistence of solutions for problem (1.1) with the sub-critical initial energy J(u₀)<d.

Theorem 4.1

(Global existence when J(u₀)<d). Let u₀ ∈ X₀, J(u₀)<d and I(u₀)>0. Then problem (1.1) admits a global weak solution u(t) ∈ L^∞(0, ∞;X₀) with u_t(t) ∈ L²(0, ∞;X₀) and u(t) ∈ W_p for 0 ⩽ t<∞.

Proof. Let ω_j(x) be a system of base functions in X₀. Construct the approximate solutions of problem (1.1) as follows

u m ( x , t ) = ∑ j = 1 m g j m ( t ) ω j ( x ) , m = 1 , 2 , ⋯

satisfying

(4.2) umt,ωs+um,ωsW0+umt,ωsW0=umq−2um,ωs,s=1,2,⋯,m

(4.3) um(x,0)=∑j=1majmωj(x)→u0(x) in X0 as m→∞.

Multiplying (4.2) by g′_sm(t), summing for s, and integrating over [0, t) in time, then

∫ 0 t ∥ u m τ ∥ X 0 2 d τ + J ( u m ) = J ( u m ( 0 ) ) , 0 ≤ t < ∞ .

By (4.3) we can get J(u_m(0)) → J(u₀), then for sufficiently large m, we have

(4.4) ∫0tumτX02dτ+Jum<d,0≤t<∞.

Next, we prove u_m(x, t) ∈ W_p for sufficiently large m and 0 ⩽ t<∞. If it is false, then there exists t₀ such that u_m(x, t₀) ∈ ∂W_p, then

I(um(t0))=0,∥um(t0)∥X0≠0 or J(um(t0))=d.

By (4.4), it implies that J(u_m(t₀))=d < J(u_m(0)) is not true. On the other hand, If I(um(t0))=0,∥um(t0)∥X0≠0 , according to the definition of d, we have J(u_m(t₀)) ⩾ d, which is also contradictive with (4.4). Hence u_m(x, t) ∈ W_p for all 0 ⩽ t<∞ and sufficiently large m. Then by (4.4) and

J(um)=1qI(um)+q−2λ2qλ∥um∥W02λ,

we obtain

(4.5) ∫0tumτX02dτ+q−2λ2qλumW02λ<d,0≤t<∞,

for sufficiently large m, which yields

∫ 0 t ∥ u m τ ∥ X 0 2 d τ < d , 0 ≤ t < ∞ .

Also, according to the embedding inequality ∥um∥2≤C∥um∥W0 , (4.5) implies

∥ u m ∥ X 0 2 λ ≤ ( 1 + C 2 ) λ ∥ u m ∥ W 0 2 λ < 2 q λ d ( 1 + C 2 ) λ q − 2 λ , 0 ≤ t < ∞ .

So, we can get

∥um∥qq≤C∗q∥um∥W0q≤C∗q∥um∥X0q<C∗q2qλd(1+C2)λq−2λq2λ,

where C_* is the embedding constant from W₀↪ L^q(Ω). Therefore, there exist a u and a subsequence u_m, such that as m → ∞.

umt⇀ut weakly in L20,∞;X0,um⇀u weakly star in L∞0,∞;X0,um⇀u weakly in L∞0,∞;Lq(Ω).

Thus in (4.2), for s fixed, letting m → ∞, then, we get

(ut,ws)+〈u,ws〉W0+(ut,ws)W0=(|u|q−2u,ws)

for all s. Further we have

(ut,v)+〈u,v〉W0+(ut,v)W0=(|u|q−2u,v), ∀v∈X0, t∈(0,∞).

Moreover, (4.3) give us u(x, 0)=u₀(x) in X₀. That means u is a global weak solution of problem (1.1). □

Theorem 4.2

(Finite time blowup when J(u₀)<d). Suppose that u₀ ∈ X₀ and u₀ ∈ V_p, then any nontrivial solution to problem (1.1) blows up in finite time. In other words, there exists a finite time T such that

(4.6) limt→T∫0t∥u∥X02dτ=+∞.

Proof. Arguing by contradiction, assume that the solution u(t) exists for all t=0. Let u(t) be any weak solution of problem (1.1) with J(u₀)<d, I(u₀)<0. For any t > 0, we define

(4.7) H(t):=∫0t∥u∥X02dτ+(T−t)u0X0.

So we can get

(4.8) H′(t)=∥u∥X02−u0X02=2∫0tu,uτX0dτ,

and

(4.9) H′′(t)=2u,utX0.

Employing the Cauchy Schwartz inequality, we obtain

(4.10) ∫0tu,uτX0dτ2≤∫0t∥u∥X02dτ∫0tuτX02dτ.

As a consequence, we read the differential inequality

(4.11) H(t)H′′(t)−q2H′(t)2=H(t)H′′(t)−2q∫0tu,uτX0dτ2≥H(t)H′′(t)−2qH(t)∫0tuτX02dτ=H(t)ξ(t),

for almost every t=0. Next we define

ξ(t):=2u,utX0−2q∫0tuτX02dτ.

Setting ϕ=u(t) in (2.4) and using (2.7), it follows that

(4.12) ξ(t)≥−2qJ(u(t))+q−2λλ∥u∥W02λ−2qJu0+2qJ(u(t))>q−2λλ∥u∥W02λ−2qd.

Then we discuss the situation in two cases.

(i) If 0 < J(u₀)<d, by (2.8) and Lemma 2.3 (ii) we have d≤qθ∗2λ−2λθ∗2λ−q2qλ∥u∥W02λ , where

θ∗=∥u∥W02λ∥u∥qq1q−2λ.

It follows lemma 4.1 that I(u) < 0 for t > 0. This implies θ_* < 1, then we can get

(4.13) d<q−2λ2qλ∥u∥W02λ.

ξ(t)>0, 0≤t≤∞,

which implies

(4.14) H(t)H′′(t)−q2H′(t)2>0,0≤t≤∞.

Further by a simple computation, it gives

(4.15) H−α(t)′′=−αHα+2(t)H(t)H′′(t)−(α+1)H′(t)2≤0,α=q−22>0.

(ii) If J(u₀)⩽0, by (4.14) and q > 2λ, we get

ξ(t)≥q−2λλ∥u∥W02λ−2qJ(u0)>0.

Obviously, we can also derive (4.14) in this case. Moreover, by J(u₀)⩽0 it implies that I(u) < 0 for all t ⩽ 0.

Then multiplying both sides of the first equation in (1.1) by u, it follows that (u,ut)X0=−I(u) , which together with (4.9) and the fact I(u) < 0 gives H″(t) > 0 for all t=0. Then by H′(0)=0, it can be deduced that H′(t) > 0 for all t > 0. Note that

(4.16) H−α(t)′=−αH′(t)Hα+1(t)<0

by H′(t) > 0 and H(t) > 0 for t > 0. From (4.15) and (4.16), it follows that there exists a finite time T > 0 such that

limt→TH−α(t)=0

and

limt→TH(t)=+∞.

Obviously, this contradicts T_max=∞. So we get

limt→T∫0t∥u∥X02dτ=+∞.

Then the proof is completed. □

4.2 Asymptotic behavior of solutions

Xu and Su in [47] studied the initial boundary value problem of semilinear pseudo-parabolic equation (1.5), obtained the asymptotic behavior of solutions with initial energy J(u₀) ⩽ d, which implies that the global solution to problem (1.5) decay exponentially. In this section, we shall consider the above decay behavior of the global solution in the fractional setting.

Theorem 4.3

(Asymptotic behavior of solutions for J(u₀)<d). Let u₀ ∈ X₀, J(u₀)<d and I(u₀)>0. Then for the global weak solution u of problem (1.1), when λ=1, there exists a constant β>0 such that

(4.17) ∥u∥X02≤u0X02e−βt,0≤t<∞,

when λ>1, then

(4.18) ∥u∥X02≤21−δ1(λ−1)t+u0X0−2(λ−1)−1λ−1,0≤t<∞,

where δ₁ is same as that in Lemma 4.1.

Proof. First, Theorem 4.1 gives the existence of global weak solutions for problem (1.1). Now we need to prove (4.17) and (4.18). Let u(t) be any global weak solution of problem (1.1) with J(u₀)<d and I(u₀)>0. Then (2.4) holds for 0 ⩽ t<∞. Multiplying (2.4) by any d(t) ∈ [0, ∞), we get

(ut,d(t)v)+〈u,d(t)v〉W0+(ut,d(t)v)W0=(|u|q−2u,d(t)v),v∈X0, d(t)∈C[0,∞)

and

(4.19) ut,w+〈u,w〉W0+ut,wW0=|u|q−2u,w.

Setting w=u, (4.19) leads to

(4.20) 12ddt∥u∥X0+I(u)=0,0≤t<∞.

From Lemma 4.1 along with 0 < J(u₀)<d and I(u₀)>0, we get u(t) ∈ W_δ for δ ∈ (δ₁, δ₂) and t ∈ [0, ∞). Hence, it follows that I_δ(u)=0 and Iδ1(u)≥0 for δ ∈ (δ₁, δ₂) and t ∈ [0, ∞). Then from (4.20) and the definition of d(δ), we have

(4.21) 12ddt∥u∥X02+1−δ1∥u∥X02λ+Iδ1(u)=0,0≤t<∞.

From (4.21) we also have

(4.22) 12ddt∥u∥X02+1−δ1∥u∥X02λ≤0,0≤t<∞.

When λ=1, by the Gronwall inequality, we have

∥u∥X02≤∥u0∥X02e−2(1−δ1)t, 0≤t<∞.

Therefore, there exists a constant β=2(1−δ₁)>0 such that

∥u∥X02≤∥u0∥X02e−βt, 0≤t<∞.

When λ>1, from (4.22) we have

ddt∥u∥X02≤−2(1−δ1)∥u∥X02λ, 0≤t<∞.

So we get

∥u∥X02≤2(1−δ1)(λ−1)t+∥u0∥X0−2(λ−1)−1λ−1, 0≤t<∞.

The proof is completed. □

4.3 Lower bound estimate of the blowup time

Luo [19] considered the semilinear pseudo-parabolic equation (1.5), obtained a lower bound for blow-up time at low initial energy. Inspiring by Luo’s work. In this section, by the similar argument, we derive the lower bound estimate for blowup time of solution to problem (1.1) with J(u₀)<d.

Theorem 4.4

(Lower bound of the blowup time when J(u₀)<d). Suppose q > 2λ, u₀ ∈ X₀, J(u₀)<d, I(u₀)<0, then the solution u(x, t) of problem blows up in finite time T in X₀-norm. Moreover,

T≥∥u0∥X0−q+2(q−2)C∗q.

Proof. First, from Theorem 4.2, we know that the solution u of problem (1.1) blows up in finite time T. Now, we estimate the lower bound for the blow-up time T. Let

(4.23) φ(t):=∥u∥X02,

multiplying u on two sides of equation (1.1), we have

(4.24) ut,u+(−Δ)sut,u=−[u]S2(λ−1)(−Δ)Su,u+|u|q−2u,u,

then by direct computation and (4.24), it follows that

(4.25) φ′(t)=−2∥u∥W02λ+2∥u∥qq.

Then by (4.25) and the embedding inequality ∥u∥q≤∥u∥X0 , it implies

φ′(t)≤2C∗q(φ(t))q2.

So we see the following inequality

(4.26) φ′(t)(φ(t))q2≤2C⋆q

Integrating the inequality (4.27) from 0 to t, we have

(4.27) (φ(0))−q−22−(φ(t))−q−22≤(q−2)C⋆qt.

So letting t → T in (4.27), we can conclude that

T≥∥u0∥X0−q+2(q−2)C∗q.

The proof is completed. □

4.4 Upper bound estimate of the blowup time

Sun et al. [33] obtained the finite time blowup results for (1.5) provided that the initial energy satisfies J(u₀)<d (∞), where d (∞) is a nonnegative constant, and also derive the estimates of the lower bound and upper bound for the blowup time. Similarly, we turn to the upper bound estimate for blowup time at sub-critical initial energy case of problem (1.1).

Theorem 4.5

(Upper bound of the blowup time when 0 < J(u₀)<d). For all q > 2λ, Assume that u₀ ∈ X₀, 0 < J(u₀)<d, I(u₀)<0, then the solution of problem (1.1) blows up in finite time. Furthermore, the maximum existence time T of u(t) satisfies

T≤4(q−1)∥u0∥X02q(q−2)2(d−J(u0)).

Proof. Since I(u₀)<0 and 0 < J(u₀)<d, by Lemma 4.1 (ii) we can get I(u (t))<0, further

(4.28) 12ddt∥u∥X02=∥u∥qq−∥u∥W02λ=−I(u(t))>0, t∈[0,T].

For T˜∈(0,T) , we define

(4.29) F(t):=∫0t∥u∥X02dτ+(T˜−t)∥u0∥X02+β(t+σ)2,

where β and σ will be give in the following prooof. Then we can write

(4.30) F′(t)=∥u∥X02−u0X02+2β(t+σ)=∫0tddτ∥u∥X02dτ+2β(t+σ)=2∫0tu,uτX0dτ+2β(t+σ)>0.

So F(t)=F (0)>0 and F(t) is strictly increasing on [0,T˜] . Furthermore, by (2.7) we can deduce

(4.31) F′′(t)=2u,utX0+2β=−2I(u(t))+2β=−2qJ(u(t))+q−2λλ∥u∥W02λ+2β=−2qJu0+2q∫0tuτX02dτ+q−2λλ∥u∥W02λ+2β.

By Cauchy-Schwartz inequality and Hölder’s inequality we can get

(4.32) ∫0tu,uτX0dτ+β(t+σ)2≤∫0t∥u∥X02dτ+β(t+σ)2∫0tuτX02dτ+β.

Therefore, in view of (4.29)-(4.32) and (4.13), we derive

(4.33) F(t)F′′(t)−q2F′(t)2=F(t)F′′(t)−2q∫0tu,uτX0dτ+β(t+σ)2≥F(t)F′′(t)−2q∫0t∥u∥X02dτ+β(t+σ)2∫0tuτX02dτ+β≥F(t)F′′(t)−2qF(t)∫0tuτX02dτ+β=F(t)−2qJu0+q−2λλ∥u∥W02λ+2β−2qβ>2qF(t)d−Ju0−(q−1)βq,

for any t∈[0,T˜] and restricting β to satisfy

(4.34) 0<β≤qq−1d−Ju0.

Let G(t):=F2−q2(t) for t∈[0,T˜] , then by F(t) > 0, F′(t) > 0, q > 2 and the above inequality, we get

(4.35) G′(t)=−q−22F−q2(t)F′(t)<0,G′′(t)=2−q2F−q+22(t)F(t)F′′(t)−q2F′(t)2<0

for all t∈[0,T˜] . Then it follows from G″(t) < 0 that

(4.36) G(T⃗)−G(0)=G′(γ)T˜<G′(0)T⃗,γ∈(0,T⃗).

By the definition of G(t), (4.29), (4.30) and (4.35), we obtain

G ( 0 ) = F 2 − q 2 ( 0 ) > 0 , G ( T ~ ) = F 2 − q 2 ( T ~ ) > 0 , G ′ ( 0 ) = 2 − q 2 F − q 2 ( 0 ) F ′ ( 0 ) = ( 2 − q ) β σ F − q 2 ( 0 ) < 0.

Combining (4.36) and the above inequalities, we can deduce

T˜≤G(T˜)G′(0)−G(0)G′(0)<−G(0)G′(0)=F(0)(q−2)βσ.

Then it follows that

T˜≤T∥u0∥X02+βσ2(q−2)βσ=σq−2+∥u0∥X02(q−2)βσT.

Hence, letting T˜→T , we get

(4.37) T≤u0X02(q−2)βσT+σq−2.

Fix any β satisfying (4.34), let σ be large enough such that

(4.38) u 0 X 0 2 ( q − 2 ) β < σ < + ∞ ,

then (4.37) leads to

(4.39) T ≤ σ q − 2 1 + u 0 X 0 2 ( 2 − q ) β σ − 1 = β σ 2 ( q − 2 ) β σ − u 0 X 0 2 .

Minimizing the last term of (4.39) for σ satisfying (4.38) one has

(4.40) T ≤ 2 u 0 X 0 2 ( q − 2 ) 2 β .

Minimizing the the last term of (4.39) for β satisfying (4.34) we finally get

(4.41) T ≤ 4 ( q − 1 ) u 0 X 0 2 q ( q − 2 ) 2 d − J u 0 .

The proof is completed. □

By the way, inspired by [19], we can also derive an upper bound for blow-up time when J(u₀)<0.

Theorem 4.6

(Upper bound of the blowup time when J(u₀)<0). If q > 2λ⩾2, u₀ ∈ X₀, J(u₀)<0, then the solution of problem (1.1) blows up at some finite time T and

T≤∥u0∥X02(2−q)qJ(u0).

Proof. If we replace the auxiliary functions φ(t) and ψ(t) used in the proof of [20, Theorem 3.1] with the following form

φ(t):=∥u∥X02,t∈[0,T]

and

ψ(t):=−2qJ(u),t∈[0,T],

which corresponds to problem (1.1). Then by similar arguments, we can derive this conclusion. □

Remark. 4.2

The above conclusion does not give the upper bound of the blowup time when J(u₀)=0. In fact, if we consider the case d(δ)=0 in Lemma 2.5 and the case e=0 in Lemma 4.1, we will find Theorem 4.5 is also valid to estimate the upper bound of blowup time when J(u₀)=0. It’s just that we don't consider the non-positive situation of d(δ), even if d(δ)=0 and e=0 satisfy the characteristics of Lemma 2.5 and Lemma 4.1, respectively.

5 Critical initial energy J(u₀)=d

5.1 Global existence and finite time blowup of solution

In this section, we prove a threshold result of global existence and nonexistence of solutions for problem (1.1) with the critical initial energy J(u₀)=d.

Theorem 5.1

(Global existence when J(u₀)=d). Assume that u₀ ∈ X₀, I(u₀)>0 and J(u₀)=d. Then the weak solution of problem (1.1) exists globally, satisfying u(t) ∈ L^∞(0, ∞;X₀) with u_t(t) ∈ L²(0, ∞;X₀) and u(t)∈Wp‾=Wp∪∂Wp for 0 ⩽ t<∞.

Proof. From J(u₀)=d, it can be deduced that ∥u0∥X0≠0 . Pick a sequence θm=1−1m , such that 0<θ_m < 1, Let u_0m(x)=θ_mu₀(x), m=2, 3, ···. Consider the initial condition u(x, 0)=u_0m(x) and the corresponding problem (1.1). From I(u₀)⩾0 and lemma 2.3 (ii), we have

θ∗=∥u0∥W02λ∥u0∥qq1q−2λ≥1.

Thus, it follows that I(u_0m)=I(θ_mu₀)>0 and J(u_0m)=J(θ_mu₀)<J(u₀)=d. Then by Theorem 4.1, it can be deduced that for each m problem (1.1) admits a global solution u_m ∈ L^∞(0, ∞;X₀) with u_mt ∈ L²(0, ∞;X₀) and u_m ∈ V_p for 0 ⩽ t<∞, satisfying

(5.1) umt,v+um,vW0+umt,vW0=umq−2um,v,∀v∈X0,t∈R0+,∫0tumτX02dτ+Jum=Ju0m<d,0≤t<∞.

Then we can get

(5.2) ∫0tumτX02dτ+q−2λ2qλumW02λ<d,0≤t<∞.

The rest of the proof is similar as that in theorem 4.1. □

Theorem 5.2

(Finite time blowup when J(u₀)=d). Suppose that u₀ ∈ X₀, I(u₀)<0 and J(u₀)=d. then any nontrivial solution to problem (1.1) must blowup in finite time T satisfying

limt→T∫0t∥u∥X02dτ=+∞.

Proof. Similar to the proof of Theorem 4.2, first we assume that the critical initial energy solution exists globally. From I(u₀)<0 and the continuity of I(u (t)) in t, it can be seen that there exists a sufficiently small tˉ>0 such that I(u (t))<0 for t∈[0,tˉ] . Moreover, by the fact that (u,ut)X0=−I(u(t))>0 for t∈[0,tˉ] , we have u_t≠0 for t∈[0,tˉ] . Hence, by (2.7) and the continuity of J(u (t)) in t, it follows that J(u (t))<d for t∈(0,tˉ] . Taking tˆ∈(0,tˉ] as the new initial time, obviously there holds I(u(tˉ))<0 and J(u(tˉ))<d . The remainder of the proof is similar to that of Theorem 4.2. □

5.2 Asymptotic behavior of solutions

In this section, we consider the asymptotic behavior of solutions for problem (1.1) with the critical initial condition J(u₀)=d. By the similar way of the proof of Theorem 4.3, we can give Theorem 5.3.

Theorem 5.3

(Asymptotic behavior of solutions for J(u₀)=d). Let u₀ ∈ X₀, J(u₀)=d and I(u₀)>0. Then for the global weak solution u of problem (1.1), when λ=1, there exists constants E > 0, t₁ > 0 and γ>0 such that

(5.3) ∥u∥X02≤Ee−γt,t1≤t<∞.

When λ>1, then

(5.4) ∥u∥X02≤21−δ1(λ−1)t−t1+ut1X0−2(λ−1)−1λ−1,t1≤t<∞.

Proof. First, Theorem 5.1 gives the existence of a global weak solution for problem (1.1). In addition, from Remark 4.1, Theorem 5.1 and (2.7), it follows that if u(t) is a global weak solution of problem (1.1) with J(u₀)=d, I(u₀)>0, we claim that J(u) < d and I(u)=0 for 0 ⩽ t<∞. Next, we consider the following two cases.

(i) Assume that I(u) > 0 for 0 ⩽ t<∞. Then from (ut,u)X0=−I(u)<0 and ∥ut∥X0>0 , it follows that ∫0t∥ut∥X02dτ is increasing on t ∈ [0, ∞). Picking any t₁ > 0 and setting

(5.5) d1=d−∫0t1uτX02dτ,

by noticing (2.7), we have 0 < J(u) ⩽ d₁<d and u(t) ∈ W_δ for δ ∈ (δ₁, δ₂) and t ∈ [t₁, ∞), where δ₁<δ₂ solve the equation d(δ)=d₁. Thus, Iδ1(u)≥0 for t=t₁, which together with (4.21), gives that

12ddt∥u∥X02+(1−δ1)∥u∥X02λ≤0, t1≤t<∞.

When λ=1, making use of Gronwall’s inequality, we can get

∥u∥X02≤∥u(t1)∥X02e−2(1−δ1)(t−t1)=∥u(t1)∥X02e2(1−δ1)t1e−2(1−δ1)t.

When λ>1, it follows that

∥u∥X02≤2(1−δ1)(λ−1)(t−t1)+∥u(t1)∥X0−2(λ−1)−1λ−1, t1≤t<∞.

(ii) Let us suppose by contradiction that t₀ > 0 is the first time such that I(u (t₀))=0. By (2.8), we get

J(u(t0))≥d.

Meanwhile, (2.7) gives

(5.6) Jut0≤d−∫0t0uτX02dτ≤d.

Hence we deduce J(u (t₀))=d. Again from (5.6) we get ∫0t0∥uτ∥X02dτ=0 , that is u(t) ≡ 0 for 0 ⩽ t ⩽ t₁, which contradicts I(u₀)>0. Hence we have I(u) > 0 and J(u) < d for 0 < t<∞.

By the continuity of the functionals J(u) and I(u) in t, we reset the initial data to a small enough t₁ > 0 such that 0 < J(u (t₁))<d and I(u (t₁))>0. By (4.21) we get

12ddt∥u∥X02+(1−δ1)∥u∥X02λ≤0, t1≤t<∞.

Making use of Gronwall’s inequality, when λ=1 we can get

∥u∥X02≤∥u(t1)∥X02e−2(1−δ1)(t−t1)=∥u(t1)∥X02e2(1−δ1)t1e−2(1−δ1)t,

when λ>1, the result is same as the case (i). Therefore, when λ=1, there exist constants E > 0, t₁ > 0 and γ>0 such that

∥u∥X02≤Ee−γt, t1≤t<∞.

The proof is completed. □

6 Blowup for arbitrary positive initial energy J(u₀)>0

In this section, we establish a finite time blowup theorem for the solution of problem (1.1) with arbitrary high initial energy. At the same time, we estimate the upper bound of the blowup time. Firstly, the invariance of the set N− is proved as follows.

Lemma 6.1

(The invariance of N− when J(u₀)>0). Assume that 2λ<q<2s∗ , u₀ ∈ X₀, J(u₀)>0 and the initial condition

(6.1) 2λq1+C2λq−2λJu0≤u0W02λ

holds. Then u∈N− for all t ∈ [0, T], where C denotes the embedding constant for W₀↪ L²(Ω), T is maximum existence time of u(t).

Proof. Let u(t) be any weak solution of problem (1.1). Multiplying (1.1) by u_t(t) and integrating on Ω, then we have

∥ut∥X02=−12λddt∥u∥W02λ+1qddt∥u∥qq,

Further we could obtain

(6.2) ddtJ(u)=−ut(t)X02≤0.

Multiplying (1.1) by u and integrate on Ω × (0, t), we have

1 2 ∥ u ∥ X 0 2 − 1 2 ∥ u 0 ∥ X 0 2 + ∫ 0 t ( ∥ u ∥ W 0 2 λ − ∥ u ∥ q q ) d τ = 0 ,

that is

(6.3) 12ddt∥u∥X02=−I(u).

Note that

J(u0)=q−2λ2λq∥u0∥W02λ+1qI(u0),

which together with (6.1) indicates that I(u₀)<0. Next, we prove u(t)∈N− for all t ∈ [0, T). Arguing by contradiction, by the continuity of I(t) in t, we assume that there exists a t˜∈(0,T) such that u(t)∈N− for 0≤t<t˜ and u(t˜)∈N , then by (6.3) we have

(6.4) ddt∥u(t)∥X02=−2I(u)>0,t∈[0,t˜),

which implies that

∥u0∥X02<∥u(t˜)∥X02.

Then, we have

(6.5) u0X02λ<∥u(t˜)∥X02λ.

From (6.2) it follows that

(6.6) J(u(t))≤Ju0 for all t∈[0,t˜].

By the definition of J(u) and u(t˜)∈N , we derive to

J(u(t˜))=q−2λ2λq∥u(t˜)∥W02λ,

which together with (6.1) and (6.6), we can get

q−2λ2λq(1+C2)λ∥u(t˜)∥X02λ≤q−2λ2λq∥u(t˜)∥W02λ≤J(u0)≤q−2λ2λq(1+C2)λ∥u0∥X02λ,

i.e., ∥u(t˜)∥X02λ<∥u0∥X02λ , which contradicts (6.5). □

Lemma 6.2

([16,17]) 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 some constant. If ψ (0)>0 and ψ′(0)>0, then there exists 0<t1≤ψ(0)θψ′(0) such that ψ(t) tends to ∞ as t → t₁.

Now we show high energy blowup and estimate the upper bound of the blowup time of solutions for problem (1.1).

Theorem 6.1

(Finite time blowup when J(u₀)>0). Let u(t) be a weak solution to problem (1.1) with u₀ ∈ X₀. Suppose that J(u₀)>0 and (6.1) holds, then the solution u(t) blows up in finite time. In addition there exists a t₁ as

0<t1≤2η(0)(α−1)η′(0),

such that

limt→t1∫0t∥u∥X02dτ=+∞,

where α, η (0) and η′(0) will be determined in the later proof.

Proof. Arguing by contradiction, we assume the existence time of solution T=+∞. Integrating of (6.2) with respect to t, we have

(6.7) J(u)+∫0tuτX02dτ=Ju0.

From (6.3) we have

(6.8) ddt∥u∥X02=−2I(u)=−2(∥u∥W02λ−∥u∥qq)=−4λ12λ∥u∥W02λ−1q∥u∥qq+2−4λq∥u∥qq=−4λJ(u)+2q−4λq∥u∥qq.

In the rest of the proof, we consider the following two cases.

(i) J(u)=0, for all t > 0. From (6.1), we choose α satisfying

(6.9) 1<α<(q−2λ)∥u0∥X022λq(1+C2)J(u0).

Substituting (6.7) into (6.8), as J(u)=0 in this case we get

(6.10) ddt∥u∥X02=4λ(α−1)J(u)−4λαJ(u)+2(q−2λ)q∥u∥qq≥−4λαJ(u0)+4λα∫0t∥uτ∥X02dτ+2(q−2λ)q∥u∥qq.

From Lemma 6.1 it follows that I(u) < 0, i.e.,

∥u∥W02λ<∥u∥qq.

Therefore, applying the basic inequality s ⩽ s^α+1 for any s=0 and α⩾1, we can obtain

(6.11) ddt∥u∥X02≥−4λαJ(u0)+4λα∫0t∥uτ∥X02dτ+2(q−2λ)q∥u∥qq>−4λαJ(u0)+4λα∫0t∥uτ∥X02dτ+2(q−2λ)q∥u∥W02λ>−4λαJ(u0)+4λα∫0t∥uτ∥X02dτ+2(q−2λ)q(∥u∥W02−1)≥−4λαJ(u0)+4λα∫0t∥uτ∥X02dτ+2(q−2λ)q(1+C2)∥u∥X02−2(q−2λ)q,

where C is the best embedding constant of inequality ∥u∥2≤C∥u∥W0 . Then

(6.12) ddt∥u∥X02−2(q−2λ)q(1+C2)∥u∥X02>−4λαJ(u0)−2(q−2λ)q,

which yields

(6.13) ∥u∥X02>∥u0∥X02e2(q−2λ)q(1+C2)t+q(1+C2)q−2λ2λαJ(u0)+q−2λq1−e2(q−2λ)q(1+C2)t.

Next, we define

y(t):=∫0t∥u(τ)∥X02dτ.

Since the solution u is global, thus the function y(t) is bounded for all t=0. Then we have

y′(t)=∥u(t)∥X02

and

y′′(t)=ddt∥u∥X02.

Substituting (6.13) into (6.11), we get

(6.14) y′′(t)>2(q−2λ)q(1+C2)∥u0∥X02−4λαJ(u0)−2(q−2λ)qe2(q−2λ)q(1+C2)t+4λα∫0t∥uτ∥X02dτ>2λαε∥u0∥X02+4λα∫0t∥uτ∥X02dτ:=A(t).

By (6.9), we can take ε>0 small enough such that

(6.15) ε<12λα∥u0∥X022(q−2λ)q(1+C2)∥u0∥X02−4λαJ(u0)−2(q−2λ)q.

Then we pick c > 0 large enough such that

(6.16) c>14ε−2∥u0∥X04.

We now define the auxiliary function

η(t):=y2(t)+ε−1∥u0∥X02y(t)+c.

Hence

(6.17) η′(t)=2y(t)+ε−1∥u0∥X02y′(t),

(6.18) η′′(t)=2y(t)+ε−1∥u0∥X02y′′(t)+2(y′(t))2.

Setting ρ:=4c−ε−2∥u0∥X04 , by (6.16) we know ρ>0. Now, from (6.17) we can write

(6.19) (η′(t))2=2y(t)+ε−1∥u0∥X022(y′(t))2=4y2(t)+4ε−1∥u0∥X02y(t)+ε−2∥u0∥X04(y′(t))2=4y2(t)+4ε−1∥u0∥X02y(t)+4c−ρ(y′(t))2=(4φ(t)−ρ)(y′(t))2.

The above equality yields

(6.20) 4η(t)(y′(t))2=(η′(t))2+ρ(y′(t))2.

By integrating the following identity from 0 to t, it gives

(6.21) 12ddt∥u(t)∥X02=(u,ut)+(u,ut)W0,

i.e.,

12∥u(t)∥X02−∥u0∥X02=∫0t(u,uτ)X0dτ.

Hence

∥u(t)∥X02=∥u0∥X02+2∫0t(u,uτ)X0dτ.

This equality along with the Hölder and Young’s inequality gives

(6.22) (y′(t))2=∥u(t)∥X04=∥u0∥X02+2∫0t(u,uτ)X0dτ2≤∥u0∥X02+2∫0t∥u∥X02dτ12∫0t∥uτ∥X02dτ122≤∥u0∥X04+4y(t)∫0t∥uτ∥X02dτ+2ε∥u0∥X02y(t)+2ε−1∥u0∥X02∫0t∥uτ∥X02dτ=:B(t).

From (6.18) and (6.20), we can get

(6.23) 2η(t)η′′(t)=22y(t)+ε−1∥u0∥X02y′′(t)+2(y′(t))2η(t)=22y(t)+ε−1∥u0∥X02y′′(t)η(t)+4(y′(t))2η(t)=22y(t)+ε−1∥u0∥X02y′′(t)η(t)+(η′(t))2+ρ(y′(t))2.

By (6.15) and the fact that e2(q−2λ)q(1+C2)>1 and η(t) > 0, we obtain

2η(t)η′′(t)−(1+α)(η′(t))2>2η(t)2y(t)+ε−1∥u0∥X024λα∫0t∥uτ∥X02dτ+2λαε∥u0∥X02−4αη(t)B(t)>4λαη(t)2y(t)+ε−1∥u0∥X022∫0t∥uτ∥X02dτ+ε∥u0∥X02−4αη(t)B(t)=4λαB(t)η(t)−4αB(t)η(t)>0,

i.e.,

η(t)η′′(t)−1+α2(η′(t))2>0, t∈[0,T],

which implies that

(η−ϵˉ(t))′′=ϵˉη(t)ϵˉ+2((ϵˉ+1)(η′(t))2−η′′(t)η(t))<0, ϵˉ=α−12>0.

Since η(0)=c>14ε−2∥u0∥X04>0 , η′(0)=ε−1∥u0∥X04>0 , by Lemma 6.2, it follows that there exists a

0<t∗≤2η(0)(α−1)η′(0),

such that

limt→t∗η−ϵˉ(t)=0,

and

limt→t∗η(t)=+∞.

As η(t) is a continuous function with respect to t, we can conclude that y(t) tends to ∞ at some t_* which contradicts T=+∞.

(ii) There exist some t˙ such that J(u(t˙))<0 .

Since J(u₀)>0, by the continuity of J(u (t)) in t, we can assume that there exists a first time t₀ > 0 such that J(u (t₀))=0 and J(u(t˙))<0 for some t˙>t0 . We take u(t˙) as a new initial datum, then from Lemma 6.1, we have u(t)∈N− for t>t˙ . Then similar to the proof of Theorem 4.2, we can prove the finite time blowup of the solution.

Combining the above two cases, we conclude that u(t) blows up in finite time. □

7 Conclusions and future works

Inspired by [34], it is natural to consider the following more general problem

(7.1) ut+M([u]s2)LKu+(−Δ)sut=|u|q−2u,in Ω×R+,u(x,0)=u0(x),in Ω,u(x,t)=0,in (RN∖Ω)×R0+,

the operator LK is given by

(7.2) LKφ(x)=∫Rn(2φ(x)−φ(x+y)+φ(x−y))K(y)dy

for every x ∈ ℝⁿ, where the kernel K:ℝ^N∖{0} → ℝ⁺ satisfies the following assumption

(H)m(x)K∈L1(RN),wherem(x)=min{|x|2,1};there existsK0>0,such thatK(x)≥K0|x|−(N+2s)for a.e.x∈RN∖{0}.

A typical example for K is the singular kernel K(x)=∣∣x∣∣^−(N+2s). In this case, up to some normalization constant, LKφ(x)=(−Δ)sφ(x) . Using the arguments similar to Sects. 4-6 of this paper, we get the existence and finite time blow up of solutions, as well as the asymptotic behavior for problem (7.1). However, the global existence for super-critical initial energy, i.e., J(u₀)>d can't be obtained because of the absence of the comparison principle. Thus, in order to prove the global well-posedness for problem (1.1) and (7.1) in the super-critical initial energy case, some new methods and strategies should be found, which will be the object of future work. At the same time, this work is helpful to analyze the observability and measurability of the control model in control system.

Acknowledgements

The authors thank the referees for their valuable remarks and comments on this paper. The first author is supported by Jiangsu key R & D plan(BE2018007).

Conflict of interest:

Authors state no conflict of interest.

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Received: 2021-06-17

Accepted: 2021-08-17

Published Online: 2021-10-04

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

Anomalous pseudo-parabolic Kirchhoff-type dynamical model

Abstract

1 Introduction and main result

2 Preliminaries

2.1 Functional spaces

Lemma 2.1

Lemma 2.2

Definition 2.1

2.2 Family of potential wells

Lemma. 2.3

Lemma 2.4

Lemma 2.5

Lemma 2.6

3 Existence and uniqueness of local solution

Lemma 3.1

Theorem 3.1

4 Sub-critical initial energy J(u0)<d

Definition 4.1

Lemma 4.1

Remark 4.1

4.1 Global existence and finite time blowup of solution

Theorem 4.1

Theorem 4.2

4.2 Asymptotic behavior of solutions

Theorem 4.3

4.3 Lower bound estimate of the blowup time

Theorem 4.4

4.4 Upper bound estimate of the blowup time

Theorem 4.5

Theorem 4.6

Remark. 4.2

5 Critical initial energy J(u0)=d

5.1 Global existence and finite time blowup of solution

Theorem 5.1

Theorem 5.2

5.2 Asymptotic behavior of solutions

Theorem 5.3

6 Blowup for arbitrary positive initial energy J(u0)>0

Lemma 6.1

Lemma 6.2

Theorem 6.1

7 Conclusions and future works

Acknowledgements

References

Journal and Issue

Articles in the same Issue

4 Sub-critical initial energy J(u₀)<d

5 Critical initial energy J(u₀)=d

6 Blowup for arbitrary positive initial energy J(u₀)>0