Existence and blow up of solutions to a Petrovsky equation with memory and nonlinear source term

Abstract

We consider the semilinear Petrovsky equation

$$u_{tt}+{\Delta }^{\mathsf{\text{2}}}u-\underset{0}{\overset{t}{\int }}g\left(t-s\right){\Delta }^{2}u\left(s\right)ds={\left|u\right|}^{p}u$$

in a bounded domain and prove the existence of weak solutions. Furthermore, we show that there are solutions under some conditions on initial data which blow up in finite time with non-positive initial energy as well as positive initial energy. Estimates of the lifespan of solutions are also given.

Mathematics Subject Classification (2000): 35L35; 35L75; 37B25.

1 Introduction

In this article, we concerned with the problem

$$\begin{array}{c}{u}_{tt}+{\Delta }^{2}u-\underset{0}{\overset{t}{\int }}g\left(t-s\right){\Delta }^{2}u\left(s\right)ds={\left|u\right|}^{p}u,x\in \Omega ,\tau >0\\ u\left(x,t\right)={\partial }_{\nu }u\left(x,t\right)=0,x\in \partial \Omega ,t\ge 0\\ u\left(x,0\right)=u_0\left(x\right),u_t\left(x,0\right)=u_1\left(x\right),x\in \Omega \end{array}$$

(1.1)

where Ω ⊂ Rⁿis a bounded domain with smooth boundary ∂Ω in order that the divergence theorem can be applied. ν is the unit normal vector pointing toward the exterior of Ω and p > 0. Here, g represents the kernel of the memory term satisfying some conditions to be specified later.

In the absence of the viscoelastic term, i.e., (g = 0), we motivate our article by presenting some results related to initial-boundary value Petrovsky problem

$$\begin{array}{c}{u}_{tt}+{{\Delta }^2}_u=f\left(u,u_t\right),x\in \Omega ,t>0\\ u\left(x,t\right)={\partial }_{\nu }u\left(x,t\right)=0,x\in \partial \Omega ,t\ge 0\\ u\left(x,0\right)=u_0\left(x\right),u_t\left(x,0\right)=u_1\left(x\right),x\in \Omega .\end{array}$$

(1.2)

Research of global existence, blow-up and energy decay of solutions for the initial boundary value problem (1.2) has attracted a lot of articles (see [1–4] and references there in).

Amroun and Benaissa [1] investigated (1.2) with f(u, u _t ) = b|u|^p-2u-h(u _t ) and proved the global existence of solutions by means of the stable set method in $H_0^2\left(\Omega \right)$ combined with the Faedo-Galerkin procedure. In [3], Messaoudi studied problem (1.2) with f(u, u _t ) = b|u|^p-2u-a|u _t |^m-2u _t . He proved the existence of a local weak solution and showed that this solution blows up in finite time with negative initial energy if p > m.

In the presence of the viscoelastic terms, Rivera et al. [5] considered the plate model:

$$u_{tt}+{\Delta }^{2}u-\underset{0}{\overset{t}{\int }}g\left(t-s\right){\Delta }^{2}u\left(s\right)ds=0$$

in a bounded domain Ω ⊂ R^Nand showed that the energy of solution decay exponentially provided the kernel function also decay exponentially. For more related results about the existence, finite time blow-up and asymptotic properties, we refer the reader to [5–16].

In the present article, we devote our study to problem (1.1). We will prove the existence of weak solutions under some appropriate assumptions on the function g and blow-up behavior of solutions. In order to obtain the existence of solutions, we use the Faedo-Galerkin method and to get the blow-up properties of solutions with non-positive and positive initial energy, we modify the method in [17]. Estimates for the blow-up time T* are also given.

2 Preliminaries

We define the energy function related with problem (1.1) is given by

$$E\left(t\right)=\frac{1}{2}\left[{∥u_t∥}^2+\left(1-\underset{0}{\overset{t}{\int }}g\left(s\right)ds\right){∥\Delta u∥}^2+\left(g\odot \Delta u\right)\left(t\right)\right]-\frac{1}{p+2}{∥u∥}_{p+2}^{p+2},$$

(2.1)

where

$$\left(g\odot v\right)\left(t\right)=\underset{0}{\overset{t}{\int }}g\left(t-s\right){∥v\left(t\right)-v\left(s\right)∥}_2^{2}ds.$$

We denote by ∥.∥ _k , the L^k-norm over Ω. In particular, the L²-norm is denoted ∥.∥₂. We use the familiar function spaces $H_0^2,H^4$ and throughout this article we assume $u_0\in H_0^2\left(\Omega \right)\cap H^4\left(\Omega \right)$ and $u_1\in H_0^2\left(\Omega \right)\cap L^2\left(\Omega \right)$.

In the sequel, we state some hypotheses and three well-known lemmas that will be needed later.

(A 1) p satisfies

$$\begin{array}{c}0<p\le \infty \left(N\le 4\right),\\ 0<p\le \frac{2\left(N-2\right)}{N-4}\left(N\ge 5\right).\end{array}$$

(A 2) g is a positive bounded C¹ function satisfying g(0) > 0, and for all t > 0

$$1-\underset{0}{\overset{\infty }{\int }}g\left(t\right)ds=l>0,$$

also there exists positive constants L₀, L₁ such that

(A 3)

$$-L_0\le g^{\prime }\left(t\right)\le 0,0\le g^{″}\left(t\right)\le L_1.$$

Lemma 1 (Sobolev-Poincare's inequality). Let p be a number that satisfies (A 1), then there is a constant C_* = C(Ω, p) such that

$${∥u∥}_p\le C_{*}{∥\Delta u∥}_2,u\in H_0^2\left(\Omega \right)$$

(2.2)

Lemma 2 [4]. Let δ > 0 and B(t) ∈ C²(0, ∞) be a nonnegative function satisfying

$$B^{″}\left(t\right)-4\left(\delta +1\right)B^{\prime }\left(t\right)+4\left(\delta +1\right)B\left(t\right)\ge 0.$$

(2.3)

If

$$B^{\prime }\left(0\right)>r_{2}B\left(0\right)+K_0,$$

(2.4)

with $r_2=2\left(\delta +1\right)-2\sqrt{\delta \left(\delta +1\right)}$, then B'(t) > K₀ for t > 0, where K₀ is a constant.

Lemma 3 [4]. If Y(t) is a non-increasing function on [t₀, ∞) and satisfies the differential inequality

$$Y^{\prime }{\left(t\right)}^2\ge a+bY{\left(t\right)}^{2+{\delta }^{-1}}fort\ge t_0\ge 0,$$

(2.5)

where a > 0, δ > 0 and b ∈ R, then there exists a finite time T* such that

$$\underset{t\to T^{*-}}{\mathsf{\text{lim}}}Y\left(t\right)=0.$$

Upper bounds for T* is estimated as follows:

(i) If b < 0, then

$$T^{*}\le t_0+\frac{1}{\sqrt{-b}}ln\frac{\sqrt{\frac{-a}{b}}}{\sqrt{\frac{-a}{b}-Y\left(t_0\right)}}.$$

(ii) If b = 0, then

$$T^{*}\le t_0+\frac{Y\left(t_0\right)}{Y^{\prime }\left(t_0\right)}.$$

(iii) If b > 0, then

$$T^{*}\le \frac{Y\left(t_0\right)}{\sqrt{a}},$$

or

$$T^{*}\le t_0+2^{\frac{3\delta +1}{2\delta }}\frac{c\delta }{\sqrt{a}}\left\{1-{\left[1+cY\left(t_0\right)\right]}^{\frac{-1}{2\delta }}\right\},$$

where $c={\left(\frac{a}{b}\right)}^{2+\frac{1}{\delta }}$.

3 Existence of solutions

In this section, we are going to obtain the existence of weak solutions to the problem (1.1) using Faedo-Galerkin's approximation.

Theorem 1 Let the assumptions (A 1)-(A 3) hold. Then there exists at least a solution u of (1.1) satisfying

$$\begin{array}{c}u\in L^{\infty }\left(0,\infty ;H_0^2\left(\Omega \right)\cap H^4\left(\Omega \right)\right),u^{\prime }\in L^{\infty }\left(0,\infty ;H_0^2\left(\Omega \right)\cap L^2\left(\Omega \right)\right),\\ u^{″}\in L^{\infty }\left(0,\infty ;L^2\left(\Omega \right)\right)\end{array}$$

(3.1)

and

$$\begin{array}{c}u\left(x,t\right)\to u_0\left(x\right)in{H}_0^2\left(\Omega \right)\cap H^4\left(\Omega \right)\\ u^{\prime }\left(x,t\right)\to u_1\left(x\right)in{H}_0^2\left(\Omega \right)\cap L^2\left(\Omega \right)\end{array}$$

as t → 0.

Proof We choose a basis {ω _k } (k = 1, 2, ...) in $H_0^2\left(\Omega \right)\cap H^4\left(\Omega \right)$ which is orthonormal in L²(Ω) and ω _k being the eigenfunctions of biharmonic operator subject to the homogeneous Dirichlet boundary condition.

Let V _m be the subspace of $H_0^2\left(\Omega \right)\cap H^4\left(\Omega \right)$ generated by the first m vectors. Define

$$u_m\left(t\right)=\sum _{k=1}^m{d}_m^k\left(t\right){\omega }_k,$$

(3.2)

where u _m (t) is the solution of the following Cauchy problem

$$\begin{array}{c}\left(u_m^{″}\left(t\right),{\omega }_k\right)+\left(\Delta u_m\left(t\right),\Delta {\omega }_k\right)-\underset{0}{\overset{t}{\int }}\left(t-s\right)\left(\Delta u_m\left(s\right),\Delta {\omega }_k\right)ds\\ -\left({\left|u_m\left(t\right)\right|}^p{u}_m\left(t\right),{\omega }_k\right)=0\forall k=1,m.\end{array}$$

(3.3)

with the initial conditions (when m → ∞)

$$\left\{\begin{array}{c}\hfill u_m\left(0\right)={\sum }_{k=1}^m\left(u_m\left(0\right),{\omega }_k\right){\omega }_k\to u_{0}in{H}_0^2\left(\Omega \right)\cap H^4\left(\Omega \right)\hfill \\ \hfill u_m^{\prime }\left(0\right)={\sum }_{k=1}^m\left(u_m^{\prime }\left(0\right),{\omega }_k\right){\omega }_k\to u_{1}in{H}_0^2\left(\Omega \right)\cap L^2\left(\Omega \right)\hfill \end{array}\right.$$

(3.4)

The approximate systems (3.3) and (3.4) are the normal one of differential equations which has a solution in [0, T _m ) for some T _m > 0. The solution can be extended to the [0, T] for any given T > 0 by the first estimate below.

First estimation. Substituting $u_m^{\prime }\left(t\right)$ instead of ω _k in (3.3), we find

$$\frac{d}{dt}\left(\frac{1}{2}{∥u_m^{\prime }∥}^2+\frac{1}{2}{∥\Delta u_m∥}^2-\frac{{∥u_m∥}_{p+2}^{p+2}}{p+2}\right)-\underset{0}{\overset{t}{\int }}g\left(t-s\right)\left(\Delta u_m\left(s\right),\Delta u_m^{\prime }\left(t\right)\right)ds=0.$$

(3.5)

Simple calculation similar to [11] yield

$$\begin{array}{l}-\underset{0}{\overset{t}{\int }}g\left(t-s\right)\left(\Delta u_m\left(s\right),\Delta u_m^{\prime }\left(t\right)\right)ds=-\underset{0}{\overset{t}{\int }}g\left(t-s\right)\underset{\Omega }{\int }\Delta u_m\left(t\right)\Delta u_m^{\prime }\left(t\right)dxds\\ -\underset{0}{\overset{t}{\int }}g\left(t-s\right)\underset{\Omega }{\int }\left(\Delta u_m\left(s\right)-\Delta u_m\left(t\right)\right)\Delta u_m^{\prime }\left(t\right)dxds\\ =\frac{1}{2}\underset{0}{\overset{t}{\int }}g\left(t-s\right)\frac{d}{dt}{∥\Delta u_m\left(s\right)-\Delta u_m\left(t\right)∥}^{2}ds-\frac{1}{2}\underset{0}{\overset{t}{\int }}g\left(t-s\right)\frac{d}{dt}{∥\Delta u_m\left(t\right)∥}^{2}ds\\ =\frac{1}{2}\frac{d}{dt}\left(g\odot \Delta u_m\right)\left(t\right)-\frac{1}{2}\left(g^{\prime }\odot \Delta u_m\right)\left(t\right)-\frac{1}{2}\frac{d}{dt}\underset{0}{\overset{t}{\int }}g\left(s\right)ds{∥\Delta u_m\left(t\right)∥}^{2}ds\\ +\frac{1}{2}g\left(t\right){∥\Delta u_m\left(t\right)∥}^2.\end{array}$$

(3.6)

Combining (3.5) and (3.6), we find

$$\begin{array}{l}\frac{d}{dt}\left(\frac{1}{2}{∥u_m^{\prime }∥}^2+\frac{1}{2}\left(1-\underset{0}{\overset{t}{\int }}g\left(s\right)ds\right){∥\Delta u_m∥}^2+\frac{1}{2}\left(g\odot \Delta u_m\right)\left(t\right)-\frac{{∥u_m∥}_{p+2}^{p+2}}{p+2}\right)\\ =\frac{1}{2}\left(g^{\prime }\odot \Delta u_m\right)\left(t\right)-\frac{1}{2}g\left(t\right){∥\Delta u_m\left(t\right)∥}^2,\end{array}$$

(3.7)

integrating (3.7) over (0, t) and using assumption (A3) we infer that

$${∥u_m^{\prime }∥}^2+{∥\Delta u_m∥}^2+\left(g\odot \Delta u_m\right)\left(t\right)-{∥u_m∥}_{p+2}^{p+2}\le C_1,$$

(3.8)

where C₁ is a positive constant depending only on ∥u₀∥, ∥u₁∥, p, and l. It follows from (3.8) that

$$\left\{\begin{array}{c}\hfill \left\{u_m\right\}isuniformlyboundedin{L}^{\infty }\left(0,T;H_0^2\left(\Omega \right)\right)\hfill \\ \hfill \left\{u_m^{\prime }\right\}isuniformlyboundedin{L}^{\infty }\left(0,T;L^2\left(\Omega \right)\right)\hfill \end{array}\right.$$

(3.9)

Second estimation. Differentiating (3.3) with respect to t, we get

$$\begin{array}{l}\left(u_m^{‴}\left(t\right),{\omega }_k\right)+\left(\Delta u_m^{\prime }\left(t\right),\Delta {\omega }_k\right)-\underset{0}{\overset{t}{\int }}{g}^{\prime }\left(t-s\right)\left(\Delta u_m\left(s\right),\Delta {\omega }_k\right)ds\\ -g\left(0\right)\left(\Delta u_m\left(t\right),\Delta {\omega }_k\right)-\left(p+1\right)\left({\left|u_m\left(t\right)\right|}^p{u}_m^{\prime }\left(t\right),{\omega }_k\right)=0.\end{array}$$

(3.10)

If we substitute $u_m^{″}\left(t\right)$ instead of ω _k in (3.10), it holds that

$$\begin{array}{l}\frac{d}{dt}\left(\frac{1}{2}{∥u_m^{″}∥}^2+\frac{1}{2}{∥\Delta u_m^{\prime }∥}^2\right)-\frac{d}{dt}\underset{0}{\overset{t}{\int }}{g}^{\prime }\left(t-s\right)\left(\Delta u_m\left(s\right),\Delta u_m^{\prime }\left(t\right)\right)ds\\ +\underset{0}{\overset{t}{\int }}{g}^{″}\left(t-s\right)\left(\Delta u_m\left(s\right),\Delta u_m^{\prime }\left(t\right)\right)ds+g^{\prime }\left(0\right)\left(\Delta u_m\left(t\right),\Delta u_m^{\prime }\left(t\right)\right)\\ -g\left(0\right)\frac{d}{dt}\left(\Delta u_m\left(t\right),\Delta u_m^{\prime }\left(t\right)\right)+g\left(0\right)\left(\Delta u_m^{\prime }\left(t\right),\Delta u_m^{\prime }\left(t\right)\right)\\ -\left(p+1\right)\left({\left|u_m\left(t\right)\right|}^p{u}_m^{\prime }\left(t\right),u_m^{″}\left(t\right)\right)=0.\end{array}$$

(3.11)

Since H²(Ω) ↪ L^2p+2(Ω), using Lemma 2, Hölder and Young's inequalities and (3.8)

$$\begin{array}{cc}\hfill \left|\left(p+1\right)\left({\left|u_m\left(t\right)\right|}^p{u}_m^{\prime }\left(t\right),u_m^{″}\left(t\right)\right)\right|& \le \left(p+1\right){∥u_m\left(t\right)∥}_{2p+2}^p.{∥u_m^{\prime }\left(t\right)∥}_{2p+2}.{∥u_m^{″}\left(t\right)∥}_2\hfill \\ \le C\left(\gamma \right){∥\Delta u_m^{\prime }\left(t\right)∥}^2+\gamma {∥u_m^{″}\left(t\right)∥}^2.\hfill \end{array}$$

(3.12)

Combining the relations (3.11), (3.12) and integrating over (0, t) for all t ∈ [0, T] with arbitrary fixed T, we obtain

$$\begin{array}{c}\frac{1}{2}{∥u_m^{″}∥}^2+\frac{1}{2}{∥\Delta u_m^{\prime }∥}^2\le \frac{1}{2}{∥u_m^{″}\left(0\right)∥}^2+\underset{0}{\overset{t}{\int }}{g}^{\prime }\left(t-s\right)\left(\Delta u_m\left(s\right),\Delta u_m^{\prime }\left(t\right)\right)ds\\ +\frac{1}{2}{∥\Delta u_m^{\prime }\left(0\right)∥}^2-\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\tau }{\int }}{g}^{″}\left(\tau -s\right)\left(\Delta u_m\left(s\right),\Delta u_m^{\prime }\left(\tau \right)\right)dsd\tau \\ -g^{\prime }\left(0\right)\underset{0}{\overset{t}{\int }}\left(\Delta u_m\left(s\right),\Delta u_m^{\prime }\left(s\right)\right)+g\left(0\right)\left(\Delta u_m\left(t\right),\Delta u_m^{\prime }\left(t\right)\right)\\ -g\left(0\right)\left(\Delta u_m\left(0\right),\Delta u_m^{\prime }\left(0\right)\right)-g\left(0\right)\underset{0}{\overset{t}{\int }}{∥\Delta u_m^{\prime }\left(s\right)∥}^{2}ds\\ +C\left(\gamma \right)\underset{0}{\overset{t}{\int }}{∥\Delta u_m^{\prime }\left(s\right)∥}^{2}ds+\gamma \underset{0}{\overset{t}{\int }}{∥u_m^{″}\left(s\right)∥}^{2}ds.\end{array}$$

(3.13)

From (3.4) and (3.8), we deduce that

$$|\frac{1}{2}{∥\Delta u_m^{\prime }\left(0\right)∥}^2-g\left(0\right)\left(\Delta u_m\left(0\right),\Delta u_m^{\prime }\left(0\right)\right)|\le L_2,$$

(3.14)

where L₂ is a positive constant independent of m. In the following, we find the upper bound for ${∥u_m^{″}\left(0\right)∥}^2$. Again we substitute $u_m^{″}\left(t\right)$ instead of ω _k in (3.3), and choosing t = 0, we arrive at

$$\left(u_m^{″}\left(0\right),u_m^{″}\left(0\right)\right)+\left(\Delta u_m\left(0\right),\Delta u_m^{″}\left(0\right)\right)-\left({\left|u_m\left(0\right)\right|}^p{u}_m\left(0\right),u_m^{″}\left(0\right)\right)=0,$$

which combined with the Green's formula imply

$${∥u_m^{″}\left(0\right)∥}^2+\left({\Delta }^2{u}_m\left(0\right),u_m^{″}\left(0\right)\right)-\left({\left|u_m\left(0\right)\right|}^p{u}_m\left(0\right),u_m^{″}\left(0\right)\right)=0.$$

(3.15)

By using (A1), (3.4) and Young's inequality, we deduce that

$$∥u_m^{″}\left(0\right)∥\le L_3,$$

(3.16)

where L₃ > 0 is a constant independent of m.

Owing to (3.8), (3.5) and Young's inequality with (A3), we deduce that

$$\begin{array}{c}\left|\underset{0}{\overset{t}{\int }}{g}^{\prime }\left(t-s\right)\left(\Delta u_m\left(s\right),\Delta u_m^{\prime }\left(t\right)\right)ds)\right|=\left|\left(\Delta u_m^{\prime }\left(t\right),\underset{0}{\overset{t}{\int }}{g}^{\prime }\left(t-s\right)\Delta u_m\left(s\right)ds\right)\right|\\ \le \gamma {∥\Delta u_m^{\prime }\left(t\right)∥}^2+\frac{1}{4\gamma }\underset{\Omega }{\int }{\left(\underset{0}{\overset{t}{\int }}{g}^{\prime }\left(t-s\right)\Delta u_m\left(s\right)ds\right)}^{2}dx\\ \le \gamma {∥\Delta u_m^{\prime }\left(t\right)∥}^2+\frac{L_0^2}{4\gamma }\underset{0}{\overset{t}{\int }}{∥\Delta u_m\left(s\right)∥}^{2}ds\\ \le \gamma {∥\Delta u_m^{\prime }\left(t\right)∥}^2+L_4\left(T\right),\end{array}$$

(3.17)

$$\begin{array}{c}\left|-\underset{0}{\overset{t}{\int }}\underset{0}{\overset{\tau }{\int }}{g}^{″}\left(\tau -s\right)\left(\Delta u_m\left(s\right),\Delta u_m^{\prime }\left(\tau \right)\right)dsd\tau \right|\\ =\underset{0}{\overset{t}{\int }}\left(\Delta u_m^{\prime }\left(\tau \right),\underset{0}{\overset{\tau }{\int }}{g}^{″}\left(\tau -s\right)\Delta u_m\left(s\right)ds\right)d\tau \\ \le \frac{1}{2}\underset{0}{\overset{t}{\int }}{∥\Delta u_m^{\prime }\left(s\right)∥}^{2}ds+\frac{1}{2}\underset{0}{\overset{t}{\int }}{\underset{\Omega }{\int }\left(\underset{0}{\overset{\tau }{\int }}{g}^{″}\left(\tau -s\right)\Delta u_m\left(s\right)ds\right)}^{2}dxd\tau \\ \le \frac{1}{2}\underset{0}{\overset{t}{\int }}{∥\Delta u_m^{\prime }\left(s\right)∥}^{2}ds+\frac{T{L}_1^2}{2}\underset{0}{\overset{t}{\int }}{∥\Delta u_m\left(s\right)∥}^{2}ds\\ \le \frac{1}{2}\underset{0}{\overset{t}{\int }}{∥\Delta u_m^{\prime }\left(s\right)∥}^{2}ds+L_5\left(T\right),\end{array}$$

(3.18)

$$\left|-g^{\prime }\left(0\right)\underset{0}{\overset{t}{\int }}\left(\Delta u_m\left(s\right),\Delta u_m^{\prime }\left(s\right)\right)ds\right|\le L_0\underset{0}{\overset{t}{\int }}{∥\Delta u_m^{\prime }\left(s\right)∥}^{2}ds+L_6\left(T\right),$$

(3.19)

and

$$\left|g\left(0\right)\left(\Delta u_m\left(t\right),\Delta u_m^{\prime }\left(t\right)\right)\right|\le \gamma {∥\Delta u_m^{\prime }\left(t\right)∥}^2+L_7\left(\gamma \right).$$

(3.20)

Now we choose γ > 0 small enough and combining (A3), (3.8), (3.13), (3.14), and (3.16)-(3.20), we get

$$\frac{1}{2}{∥u_m^{″}∥}^2+\frac{1}{2}{∥\Delta u_m^{\prime }∥}^2\le L_8\left(\underset{0}{\overset{t}{\int }}{∥u_m^{″}\left(s\right)∥}^{2}ds+\underset{0}{\overset{t}{\int }}{∥\Delta u_m^{\prime }\left(s\right)∥}^{2}ds\right)+L_9.$$

(3.21)

By using Gronwall's lemma we arrive at

$$\frac{1}{2}{∥u_m^{″}∥}^2+\frac{1}{2}{∥\Delta u_m^{\prime }∥}^2\le L_{10},$$

(3.22)

for all t ∈ [0, T], and L₁₀ is a positive constant independent of m. Estimate (3.22) implies

$$\left\{\begin{array}{c}\hfill \left\{u_m^{″}\right\}isuniformlyboundedin{L}^{\infty }\left(0,T;L^2\left(\Omega \right)\right)\hfill \\ \hfill \left\{u_m^{\prime }\right\}isuniformlyboundedin{L}^{\infty }\left(0,T;H_0^2\left(\Omega \right)\right)\hfill \end{array}\right.$$

(3.23)

By attention to (3.9) and (3.23), there exists a subsequence {u _i } of {u _m } and a function u such that

$$\left\{\begin{array}{c}\hfill u_i\rightharpoonup uweaklystarin{L}^{\infty }\left(0,T;H_0^2\left(\Omega \right)\right)\hfill \\ \hfill u_i^{\prime }\rightharpoonup u^{\prime }weaklystarin{L}^{\infty }\left(0,T;H_0^2\left(\Omega \right)\right)\hfill \\ \hfill u_i^{″}\rightharpoonup u^{″}weaklystarin{L}^{\infty }(0,T;L^2\left(\Omega \right))\hfill \end{array}\right.$$

(3.24)

By Aubin-Lions compactness lemma [18], it follows from (3.24) that

$$\left\{\begin{array}{c}\hfill u_i\to ustronglyinC\left(\left[0,T\right];H_0^2\left(\Omega \right)\right)\hfill \\ \hfill u_i^{\prime }\to ustronglyinC\left(\left[0,T\right]\right);L^2\left(\Omega \right))\hfill \end{array}\right.$$

(3.25)

In the sequel we will deal with the nonlinear term. By (3.9) and Sobolev embedding theorem, we obtain

$$\left\{{\left|u_m\right|}^p{u}_m\right\}isuniformlyboundedin{L}^{\infty }\left(0,T;L^2\left(\Omega \right)\right),$$

(3.26)

and therefore we can extract a subsequence {u _i } of {u _m } such that

$${\left|u_i\right|}^p{u}_i\rightharpoonup {\left|u\right|}^{p}uweaklystarin{L}^{\infty }\left(0,T;L^2\left(\Omega \right)\right).$$

(3.27)

Applying (3.24), (3.27) and letting i → ∞ in (3.3), we see that u satisfies the equation. For the initial conditions by using (3.4), (3.25) and the simple inequality

$${∥u-u_0∥}_{H_0^2\left(\Omega \right)}\le {∥u-u_i∥}_{H_0^2\left(\Omega \right)}+{∥u_i-u_i\left(0\right)∥}_{H_0^2\left(\Omega \right)}+{∥u_i\left(0\right)-u_0∥}_{H_0^2\left(\Omega \right)},$$

we get the first initial condition immediately. In the similar way, we can show the second initial condition and the proof is complete.

4 Blow-up of solutions

In this section, we study blow-up property of solutions with non-positive initial energy as well as positive initial energy, and estimate the lifespan of solutions. For this purpose, we assume that g is positive and C¹ function satisfying

(A 4)

$$g\left(0\right)>0,g^{\prime }\left(s\right)\le 0,1-\underset{0}{\overset{\infty }{\int }}g\left(s\right)ds=l>0,$$

and we make the following extra assumption on g

(A 5)

$$\underset{0}{\overset{\infty }{\int }}g\left(s\right)ds<\frac{p}{1+p}.$$

From (2.1), (A4) and Lemma 1, we have

$$\begin{array}{c}E\left(t\right)\ge \frac{1}{2}\left[\left(1-\underset{0}{\overset{t}{\int }}g\left(s\right)ds\right){∥\Delta u∥}^2+\left(g\odot \Delta u\right)\left(t\right)\right]-\frac{1}{p+2}{∥u∥}_{p+2}^{p+2}\\ \ge \frac{1}{2}\left[l{∥\Delta u∥}^2+g\odot \Delta u)\left(t\right)\right]-\frac{C_1^{p+2}{l}^{\frac{p+2}{2}}}{p+2}{∥\Delta u∥}^{p+2}\\ \ge G\left(\sqrt{l{∥\Delta u∥}^2+\left(g\odot \Delta u\right)\left(t\right)}\right),t\ge 0,\end{array}$$

where $G\left(\lambda \right)=\frac{1}{2}{\lambda }^2-\frac{C_1^{p+2}}{p+2}{\lambda }^{p+2},C_1=\frac{C_{*}}{\sqrt{l}}$. It is easy to verify that G(λ) has a maximum at ${\lambda }_1=C_1^{-\frac{p+2}{p}}$ and the maximum value is $E_1=\frac{p}{2p+4}{C}_1^{-\frac{2p+4}{p}}$.

Lemma 4 Let (A4) hold andu be a local solution of (1.1). Then E(t) is a non-increasing function on [0, T] and

$$\frac{d}{dt}E\left(t\right)=\frac{1}{2}\left(g^{\prime }\odot \Delta u\right)\left(t\right)-\frac{1}{2}g\left(t\right){∥\Delta u∥}^2\le 0,$$

(4.2)

for almost every t ∈ [0, T].

Proof Multiplying (1.1) by u _t , integrating over Ω, and finally integrating by parts, we obtain (4.2) for any regular solution. Then by density arguments, we have the result.

Lemma 5 Let (A4) hold and u be a local solution of (1.1) with initial data satisfying E(0) < E₁ and $l^{\frac{1}{2}}∥\Delta u_0∥>{\lambda }_1$. Then there exists λ₂ > λ₁ such that

$$l{∥\Delta u∥}^2+\left(g\odot \Delta u\right)\left(t\right)\ge {\lambda }_2^2.$$

(4.3)

Proof See Li and Tsai [11].

The choice of the functional is standard (see [19])

$$\psi \left(t\right)={∥u∥}^2.$$

(4.4)

It is clear that

$${\psi }^{\prime }\left(t\right)=2\left(u,u_t\right),$$

(4.5)

and from (1.1)

$${\psi }^{″}\left(t\right)=2{∥u_t∥}^2-2{∥\Delta u∥}^2+2{∥u∥}_{p+2}^{p+2}+2\underset{0}{\overset{t}{\int }}g\left(t-s\right)\left(\Delta u\left(t\right),\Delta u\left(s\right)\right)ds.$$

(4.6)

Lemma 6 Let u be a solution of (1.1) and (A4), (A5) hold, then we have

$${\psi }^{″}\left(t\right)-\left(4+p\right)\underset{\Omega }{\int }{u}_t^{2}dx\ge m\left(l{∥\Delta u∥}^2+\left(g\odot \Delta u\right)\left(t\right)\right)-\left(4+2p\right)E\left(0\right),$$

(4.7)

where $m=\left(1+p\right)-\frac{1}{l}>0$.

Proof Using the Hölder and Young's inequalities, we arrive at

$$\begin{array}{c}\underset{0}{\overset{t}{\int }}g\left(t-s\right)\left(\Delta u\left(t\right),\Delta u\left(s\right)\right)ds\ge -\left[\frac{1}{2}\left(g\odot \Delta u\right)\left(t\right)+\frac{1}{2}\underset{0}{\overset{t}{\int }}g\left(s\right)ds{∥\Delta u∥}^2\right]\\ +\underset{0}{\overset{t}{\int }}g\left(s\right)ds{∥\Delta u∥}^2,\end{array}$$

therefore (4.6) becomes

$$\begin{array}{c}{\psi }^{″}\left(t\right)-\left(4+p\right){∥u_t∥}^2\ge -\left(2+p\right){∥u_t∥}^2-2{∥\Delta u∥}^2-\left(g\odot \Delta u\right)\left(t\right)\\ +\underset{0}{\overset{t}{\int }}g\left(s\right)ds{∥\Delta u∥}^2+2{∥u∥}_{p+2}^{p+2}.\end{array}$$

Then, using (4.2), we obtain

$$\begin{array}{c}{\psi }^{″}\left(t\right)-\left(4+p\right){∥u_t∥}^2\ge -\left(4+2p\right)E\left(0\right)+p{∥\Delta u∥}^2+\left(1+p\right)\left(g\odot \Delta u\right)\left(t\right)\\ -\left(1+p\right)\underset{0}{\overset{t}{\int }}g\left(s\right)ds{∥\Delta u∥}^2-\left(2+p\right)\underset{0}{\overset{t}{\int }}\left(g^{\prime }\odot \Delta u\right)\left(s\right)ds,\end{array}$$

and so by (2.5) and (A5), we deduce

$$\begin{array}{c}{\psi }^{″}\left(t\right)-\left(4+p\right){∥u_t∥}^2\ge -\left(4+2p\right)E\left(0\right)+\left(p-\left(1+p\right)\left(1-l\right)\right){∥\Delta u∥}^2\\ +\left(1+p\right)\left(g\odot \Delta u\right)\left(t\right),\end{array}$$

(4.8)

if we set $m:=\left(1+p\right)-\frac{1}{l}$ then inequality (4.8) yields the desired result.

Consequently, we have the following result.

Lemma 7 Assume that (A4) and (A5) hold. u be a local solution of (1.1) and that either one of the following four conditions is satisfied:

(i) E(0) < 0

(ii) E(0) = 0 and ψ'(0) > 0

(iii) $0<E\left(0\right)<\frac{m}{p}{E}_1$ and $l^{\frac{1}{2}}∥\Delta u_0∥>{\lambda }_1$

(iv) $\frac{m}{p}{E}_1\le E\left(0\right)$ and ${\psi }^{\prime }\left(0\right)>r_2\left[\psi \left(0\right)+\frac{\left(4+2p\right)E\left(0\right)}{4+p}\right]$.

Then ψ' (t) > 0 for t > t*, where

in case (i)

$$t^{*}=\mathsf{\text{max}}\left\{0,\frac{{\psi }^{\prime }\left(0\right)}{\left(4+2p\right)E\left(0\right)}\right\},$$

(4.9)

in cases (ii), (iv)

$$t^{*}=0,$$

(4.10)

and in case (iii)

$$t^{*}=\mathsf{\text{max}}\left\{0,\frac{-{\psi }^{\prime }\left(0\right)}{\left(4+2p\right)\left(\frac{m}{p}{E}_1-E\left(0\right)\right)}\right\}.$$

(4.11)

Proof Suppose that condition (i) is satisfied. Then from (4.5), we have

$${\psi }^{\prime }\left(t\right)\ge {\psi }^{\prime }\left(0\right)-\left(4+2p\right)E\left(0\right)t,t\ge 0.$$

Thus ψ'(t) > 0 for t > t*, and it is easy to see that t* satisfies (4.9).

If E(0) = 0, then by using (4.3) we have ψ" (t) ≥ 0, and since ψ'(0) > 0 we arrive at

$${\psi }^{\prime }\left(t\right)>0,fort>0.$$

If $0<E\left(0\right)<\frac{m}{p}{E}_1$ and $l^{\frac{1}{2}}∥\Delta u_0∥>{\lambda }_1$ then by Lemma 4, we see that

$$\begin{array}{c}m\left(l{∥\Delta u∥}^2+\left(g\odot \Delta u\right)\left(t\right)\right)-\left(4+2p\right)E\left(0\right)\ge m{\lambda }_2^2-\left(4+2p\right)E\left(0\right)\\ >m\frac{4+2p}{p}{E}_1-\left(4+2p\right)E\left(0\right)\\ =\left(4+2p\right)\left[\frac{m}{p}{E}_1-E\left(0\right)\right].\end{array}$$

Thus from (4.5), we have

$$\begin{array}{c}{\psi }^{″}\left(t\right)\ge m\left(l{∥\Delta u∥}^2+\left(g\odot \Delta u\right)\left(t\right)\right)-\left(4+2p\right)E\left(0\right)\\ >\left(4+2p\right)\left[\frac{m}{p}{E}_1-E\left(0\right)\right]>0,\end{array}$$

(4.12)

and integrating (4.12) from 0 to t gives

$${\psi }^{\prime }\left(t\right)>0,fort\ge t^{*},$$

where t* satisfies (4.11).

Let $\frac{m}{p}{E}_1\le E\left(0\right)$, this assumption causes that

$${\psi }^{″}\left(t\right)-\left(4+p\right){∥u_t∥}^2+\left(4+2p\right)E\left(0\right)\ge 0,$$

and by using Hölder and Young's inequalities, we get

$${∥u_t∥}^2\ge {\psi }^{\prime }\left(t\right)-\psi \left(t\right),$$

thus

$${\psi }^{″}\left(t\right)-\left(4+p\right){\psi }^{\prime }\left(t\right)+\left(4+p\right)\psi \left(t\right)+\left(4+2p\right)E\left(0\right)\ge 0.$$

(4.13)

We see that the hypotheses of Lemma 2 are fulfilled with

$$\delta =\frac{p}{4}andB\left(t\right)=\psi \left(t\right)+\frac{\left(4+2p\right)E\left(0\right)}{4+p}$$

and the conclusion of Lemma 2.2 gives us

$${\psi }^{\prime }\left(t\right)>0,fort>0.$$

Therefore the proof is complete.

To estimate the life-span of ψ(t), we define the following functional

$$Y\left(t\right)=\psi {\left(t\right)}^{-\frac{p}{4}},fort\ge 0.$$

(4.14)

Then we have

$$Y^{\prime }\left(t\right)=\frac{p}{4}Y{\left(t\right)}^{1+\frac{4}{p}}{\psi }^{\prime }\left(t\right),$$

(4.15)

$$Y^{″}\left(t\right)=-\frac{p}{4}Y{\left(t\right)}^{1+\frac{8}{p}}\left[{\psi }^{″}\left(t\right)\psi \left(t\right)-\left(1+\frac{p}{4}\right){\left({\psi }^{\prime }\left(t\right)\right)}^2\right].$$

(4.16)

Using (4.4)-(4.6) and exploiting Holder's inequality on ψ'(t), we get

$$\begin{array}{c}{\psi }^{″}\left(t\right)\psi \left(t\right)-\left(1+\frac{p}{4}\right){\left({\psi }^{\prime }\left(t\right)\right)}^2\\ \ge \left[\left(l{∥\Delta u∥}^2+\left(g\odot \Delta u\right)\left(t\right)\right)-\left(4+2p\right)E\left(0\right)+\left(4+p\right){∥u_t∥}^2\right]\psi \left(t\right)\\ -4\left(1+\frac{p}{4}\right){∥u_t∥}^2\psi \left(t\right)\\ =\left[\left(l{∥\Delta u∥}^2+\left(g\odot \Delta u\right)\left(t\right)\right)-\left(4+2p\right)E\left(0\right)\right]Y{\left(t\right)}^{\frac{-4}{p}}.\end{array}$$

Utilizing the last inequality into (4.16) yields

$$Y^{″}\left(t\right)\le -\frac{p}{4}\left[\left(l{∥\Delta u∥}^2+\left(g\odot \Delta u\right)\left(t\right)\right)-\left(4+2p\right)E\left(0\right)\right]Y{\left(t\right)}^{1+\frac{4}{p}}.$$

(4.17)

Now we should assume different values for initial energy E(0).

• (1) At first if E(0) ≤ 0 then from (4.17) we have

  $$Y^{″}\left(t\right)\le \frac{p}{4}\left(4+2p\right)E\left(0\right)Y{\left(t\right)}^{1+\frac{4}{p}},$$

  (4.18)

on the other hand by Lemma 7, Y'(t) < 0 for t > t*. Multiplying (4.18) by Y'(t) and integrating from t* to t, we deduce that

$$Y^{\prime }{\left(t\right)}^2\ge \alpha +\beta Y{\left(t\right)}^{2+\frac{4}{p}}fort\ge t^{*},$$

where

$$\alpha =\frac{p^2}{16}Y{\left(t^{*}\right)}^{2+\frac{8}{p}}\left[{\psi }^{\prime }{\left(t^{*}\right)}^2-8E\left(0\right)Y{\left(t^{*}\right)}^{-\frac{4}{p}}\right]>0,$$

(4.19)

and

$$\beta =\frac{p^2}{2}E\left(0\right).$$

(4.20)

Then the hypotheses of Lemma 3 are fulfilled with $\delta =\frac{p}{4},t_0=t^{*}$ and using the conclusion of Lemma 3, there exists a finite time T* such that ${\mathsf{\text{lim}}}_{t\to T^{*-}}Y\left(t\right)=0$, i.e., in this case some solutions blow up in finite time T*.

• (2) If $0<E\left(0\right)<\frac{m}{p}{E}_1$, then from (4.17) and (4.12) we have

  $$Y^{″}\left(t\right)\le -\frac{p}{4}\left(4+2p\right)\left[\frac{m}{p}{E}_1-E\left(0\right)\right]Y{\left(t\right)}^{1+\frac{4}{p}}.$$

Then using the same arguments as in (1), we get

$$Y^{\prime }{\left(t\right)}^2\ge {\alpha }_1+{\beta }_{1}Y{\left(t\right)}^{2+\frac{4}{p}}fort\ge t^{*},$$

where

$${\alpha }_1=\frac{p^2}{16}Y{\left(t^{*}\right)}^{2+\frac{8}{p}}\left({\psi }^{\prime }{\left(t^{*}\right)}^2+8\left[\frac{m}{p}{E}_1-E\left(0\right)\right]Y{\left(t^{*}\right)}^{-\frac{4}{p}}\right)>0,$$

(4.21)

and

$${\beta }_1=\frac{p^2}{2}\left[E\left(0\right)-\frac{m}{p}{E}_1\right].$$

(4.22)

Thus by Lemma 3, there exists a finite time T* such that

$$\underset{t\to T*-}{\mathsf{\text{lim}}}\psi \left(t\right)=\infty .$$

• (3) $\frac{m}{p}{E}_1\le E\left(0\right)$. In this case, it is easy to see that by using (4.19) and (4.20) into discussion in part (1), we obtain

  $$\alpha >0ifandonlyifE\left(0\right)<\frac{{\psi }^{\prime }{\left(t^{*}\right)}^2}{8\psi \left(t^{*}\right)}.$$

Hence, Lemma 3 yields the blow-up property in this case.

Therefore, we proved the following theorem.

Theorem 2 Assume that (A4) and (A5) hold. u be a local solution of (1.1) and that either one of the following four conditions is satisfied:

(i) E(0) < 0

(ii) E(0) = 0 and ψ'(0) > 0

(iii) $0<E\left(0\right)<\frac{m}{p}{E}_1$ and $l^{\frac{1}{2}}∥\Delta u_0∥>{\lambda }_1$

(iv) $\frac{m}{p}{E}_1\le E\left(0\right)$ and ${\psi }^{\prime }\left(0\right)>r_2\left[\psi \left(0\right)+\frac{\left(4+2p\right)E\left(0\right)}{4+p}\right]$ holds.

Then the solution u blows up at finite time T*. Moreover, the upper bounds for T* can be estimated according to the sign of E(0):

in case (i)

$$T^{*}\le t^{*}-\frac{Y\left(t^{*}\right)}{Y^{\prime }\left(t^{*}\right)}.$$

Furthermore, if $Y\left(t^{*}\right)<\mathsf{\text{min}}\left\{1,\sqrt{\frac{\alpha }{-\beta }}\right\}$, then

$$T^{*}\le t^{*}+\frac{1}{\sqrt{-\beta }}ln\frac{\sqrt{\frac{\alpha }{-\beta }}}{\sqrt{\frac{\alpha }{-\beta }}-Y\left(t^{*}\right)}$$

in cases (ii)

$$T^{*}\le t^{*}-\frac{Y\left(t^{*}\right)}{Y^{\prime }\left(t^{*}\right)}or{T}^{*}\le t^{*}+\frac{Y\left(t^{*}\right)}{\sqrt{\alpha }}$$

in case (iii)

$$T^{*}\le t^{*}-\frac{Y\left(t^{*}\right)}{Y^{\prime }\left(t^{*}\right)}.$$

Furthermore, if $Y\left(t^{*}\right)<\mathsf{\text{min}}\left\{1,\sqrt{\frac{\alpha }{-\beta }}\right\}$, then

$$T^{*}\le t^{*}+\frac{1}{\sqrt{-{\beta }_1}}ln\frac{\sqrt{\frac{{\alpha }_1}{-{\beta }_1}}}{\sqrt{\frac{{\alpha }_1}{-{\beta }_1}}-Y\left(t^{*}\right)}$$

and in case (iv)

$$T^{*}\le \frac{Y\left(t^{*}\right)}{\sqrt{\alpha }}or{T}^{*}\le t^{*}+2^{\frac{3p+4}{2p}}\frac{pc}{4\sqrt{\alpha }}\left[1-{\left(1+cY\left(t^{*}\right)\right)}^{\frac{-2}{p}}\right],$$

where $d={\left(\frac{\beta }{\alpha }\right)}^{\frac{p}{p+8}}$. Here α, β, α₁, and β₁ are given in (4.19)-(4.22), respectively. Note that each t* in the above cases satisfy the same case in Lemma 7.