A class of fourth-order hyperbolic equations with strongly damped and nonlinear logarithmic terms

1.   Introduction

In this paper, we study the following initial boundary value problem:

where $\Omega\subset R^n$ is a bound domain with smooth boundary $\partial\Omega$, the vector $n$ is the unit outer normal to $\partial\Omega$ and $k$ is a positive real number. $p$ satisfies

The differential equations studied by many researchers are significant [3,8,36], especially the logarithmic nonlinear problem. The logarithmic nonlinear problem is applied to many branches of physics, such as nuclear physics, optics, and geophysics [5,6,18], and it appears naturally in inflation cosmology and supersymmetric filed theories, quantum mechanics and nuclear physics [4,15]. Fourth-order differential equation with strong damping term has wide application in viscoelastic mechanics and quantum mechanics [7,10,34]. The strong damping term $\Delta u_t$ indicates that the stress is proportional not only to the strain in the Hook law, but also to the strain rate in the linearized Kelvin-Voigt material.

Górka [18] studied the one-dimensional Klein-Gordon equation with logarithmic source terms

by using Galerkin's method, logarithmic Sobolev inequality and compactness theorem, the existence of weak solutions is obtained. Gazenave and Haraux [9] considered the problem

they prove the existence and uniqueness of weak solutions in three dimensions. Later, in the case of infinite dimension in reference [25], Lian et al. modified the potential well method and combined with Sobolev inequality to obtain the global existence of the solution and the blow up result under the condition of different initial energy($E(0)<d$, $E(0) = d$, $E(0)>d$). When $u\ln {\left| u \right|^k}$ in problem (3) becomes ${\left| u \right|^p}\ln \left| u \right|$, the problem is also considered by Lian et al.[26], and they establish the global existence and finite time blow up of solutions at three different energy levels.

Hiramatus et al.[21] introduced the equation

to study the dynamics of Q-ball in theoretical physics. A numerical research was mainly carried out, but there was no theoretical research in that paper. For problem (4), Han [19] obtained the result of global existence of weak solution in three-dimensional bounded domain, and Zhang et al.[42] proved the energy decay estimate in infinite dimension case.

Al-Gharabli and Messaoudi [1] considered the Neumann problem of weakly damped wave equations with logarithmic source term

in two-dimensional bounded domain, they first obtained the existence of weak solutions by Galerkin method. Secondly, under the framework of potential well, they proved the global existence and exponential decay of weak solutions for all the conditions where$({u_0},{u_1}) \in H_0^2 \times {L^2}$ satisfy $I({u_0}) > 0$ and $0 < E(0) < d$. In reference [2], $h({u_t})$ replaces $u_t$ in equation (5), Al-Gharabli and Messaoudi proved the existence and energy decay of solutions in the two-dimensional case.

For hyperbolic equations with nonlinear damping terms, there have also been extensive studies in recent years. Messaoudi [33] studied the following equation with nonlinear damping terms and polynomial source terms

where $a,b>0$, $m,p>2$. First, the author obtained the local existence of the solution by the contraction mapping principle. In addition, the global existence of the solution is proved under the condition of $m\geq p$, and the blow up results are obtained under the condition of $m<p$ and negative initial energy. After that, Wu and Tsai [38] generalized the result of [33]. Chen and Zhou [13] further studied the problem (6) and proved the global nonexistence of the solution with positive initial energy. Moreover, in the case of linear damping $(m = 2)$, they obtained blow up result of the solution even if the initial energy disappears under certain conditions.

Liu [31] considered the equation

When $2\leq m<p$ and the initial energy $E(0)<d$, the author obtained the global existence of weak solution and decay estimate. When the initial energy is negative, the author proved the blow-up results at finite time.

Gazzola and Squassina [17] studied the following damped semilinear wave equation

For the initial energy $E(0)<d$, the authors obtained the global existence of the weak solution. In addition, when $\omega = 0$, they obtained the finite time blow up result at any arbitrarily high initial energy.

On the basis of reference [17], Lian and Xu [28] studied the following semilinear wave equation with logarithmic source term

the author studied the global existence, asymptotic behavior and the blow up results under the conditions of subcritical initial energy and critical initial energy respectively. Under the condition of $\omega = 0$ and $E(0)>0$, the author obtained the blow up results at infinite time.

Recently, Yang et al.[40] investigated a class of fourth order strongly damped nonlinear wave equations

they comprehensively investigated the global existence, long-time behavior and finite time blow up of the solution at three different initial energy levels. Zeng and Zhao [41] considered the Cauchy problem of a Keller-Segel type chemotaxis model with logarithmic sensitivity and logistic growth, and obtain similar results. In [14], Di considered the initial boundary value problem of the fourth order wave equation with an internal nonlinear source ${\left| u \right|^\rho }u$, they proved the global existence and uniqueness of the regular solution and the weak solution respectively, and studied the explicit decay rate estimation of energy. Liu and Zhou [32] considered the local well-posedness of solutions to the initial boundary value problem for fourth-order plate equations with Hardy-Hénon potential and polynomial nonlinearity, and also studied the global existence and finite time blow-up results of solutions.

After looking up these literatures on the dynamic behavior of logarithmic term, it is not difficult to find that the estimates of power-type nonlinear term cannot be directly generalized to logarithmic nonlinear term. When the logarithmic source term is $u\ln \left| u \right|$, the logarithmic Sobolev inequality is usually used to deal with such problem. In this paper, the non-linear logarithmic source term ${\left| u \right|^{p - 2}}u\log {\left| u \right|^k}$ brings us some difficulties, here we cannot apply logarithmic Sobolev inequality. In addition, the hyperbolic equation is different from the parabolic equation. The parabolic equation with strong damping term has been extensively studied by many authors, and a large number of results have been obtained [11,20,27,29,37,39,43]. Specially, Chen and Xu [12] studied the initial boundary value problem of infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinear terms, and obtained the global existence, blow up and the asymptotic behavior of the solution. However, in this paper, for the fourth order hyperbolic equation, the emergence of logarithmic term and strong damping term prevent us from obtaining the blow up result of the solution. Here, we use the potential well method and some new techniques to obtain the existence of solution, estimate of energy decay, and blow up results.

This paper is organized as follows: In Section 2, we introduce some mathematical symbols, basic definitions and important lemmas needed for theorem proof. In Section 3, we prove the local existence of the weak solution of the problem (1). In Section 4, we give the global existence of the solution and energy decay. In the last Section, we obtain the result of the blow up at a finite time.

2.   Preliminaries and main results

In this section, we first introduce some of the notation used in this paper. The norm of $L^p(\Omega)$ is denoted by $\|\cdot\|_p$, where $1\leq p\leq\infty$. We define the following space for further discussion

Naturally, by Poincaré's inequality [30], $\|\Delta \cdot\|_2$ is the equivalent norm of $\|\cdot\|_{\mathrm{H}}$. Besides, $\langle\cdot , \cdot\rangle$ represents the duality pairing between $H^{-2}(\Omega)$ and $\mathrm{H}$.

Let us introduce some of the required functionals.

By (8) and (9), we have

We define the Nehari manifold

The depth of potential well is defined as

by Lemma 2.8, we know that $d$ satisfies

where $r_*$ is the positive constant defined in Lemma 2.7.

The potential well (stable set) $W$ and the outer space of potential well (unstable set) $V$ are defined as follows:

Now, we give the definition of weak solution to problem(1).

Definition 2.1. The function $u = u(x,t)$ is called a weak solution of the problem(1) on $\Omega \times \left[ {0,T} \right)$, if $u \in C\left( {\left[ {0,T} \right],\mathrm H} \right) \cap {C^{\rm{1}}}\left( {\left[ {0,T} \right],{L^2}\left( \Omega \right)} \right) \cap {C^{\rm{2}}}\left( {\left[ {0,T} \right],{H^{ - {\rm{2}}}}(\Omega )} \right)$, $u_t\in L^2(0,T;H_0^1(\Omega))$, and there holds

for any $v\in \mathrm H$, $t\in[0,T)$, where $u\left( {x,0} \right) = {u_0}\left( x \right)$ in $\mathrm H$, ${u_t}\left( {x,0} \right) = {u_1}\left( x \right)$ in $L^2(\Omega)$.

Next, we state our main results of this paper as follows:

Theorem 2.2. (Local existence) Suppose that $u_0\in \mathrm H$, $u_1(x)\in L^2(\Omega)$. Then there exist a $T>0$ such that problem(1) admits a unique weak solution $u$ on $[0,T]$ satisfying

Theorem 2.3. (Global existence and decay estimate) Assume (2) holds, if $u_0\in \mathrm H$, $u_1\in L^2(\Omega)$, $E(0)\leq M$ and $I(u_0)\geq0$, then the problem(1) admits a global weak solution $u \in {L^\infty }(0,\infty; \mathrm H)$ with ${u_t} \in {L^2}\left( {0,\infty;H_0^1\left( \Omega \right)} \right)$. Furthermore, there exists a positive constant ${ \mathcal K_0}$ such that the energy functional $E\left( t \right)$ satisfies the following polynomial decay estimate:

In particular, if $E\left( 0 \right) < \min \left\{ {M,\frac{{p - 2}}{{2p}}{{\left( {\frac{{e\mu }}{{k{C_2}^{p + \mu }}}} \right)}^{\frac{2}{{p + \mu - 2}}}}} \right\}$ and $0<\mu<2^*-p$, then there exist positive constants ${\mathcal K_1}$ and ${\mathcal K_2}$, such that the $E\left( t \right)$ satisfies the exponential decay estimate as follows:

where $M$ is the positive constant defined in (12).

Theorem 2.4. (Blow up) Assume $u_0\in \mathrm H \backslash \{0\}$, ${u_1} \in {L^2}(\Omega)$ satisfy $E\left( 0 \right) < d$, $I\left( {{u_0}} \right) < 0$, then the solution $u$ of problem (1) blows up in finite time.

To prove our main results, we need to introduce some lemmas.

Let (2) holds, by Sobolev's embedding theorem [16], we know that ${\left\| u \right\|_p} \le {C_p}{\left\| {\Delta u} \right\|_2},$ where $C_p$ is the optimal embedding constant of $\mathrm{H}\hookrightarrow L^p(\Omega)$, i.e. ${C_p} = \mathop {\sup }\limits_{u \in \mathrm H\backslash \{ 0\} } \frac{{{{\left\| u \right\|}_p}}}{{{{\left\| {\Delta u} \right\|}_2}}}.$ We define

for any $\alpha \in \left[ {0,{\alpha ^*}} \right)$, then $\mathrm{H}\hookrightarrow L^{p+\alpha}(\Omega)$. And we denote $C_{p+\alpha}$ by $C_*$.

Lemma 2.5. Let $u\in \mathrm{H}\setminus\{0\}$, then

$\mathrm{(1)} \lim\limits_{\lambda\rightarrow0^+}J(\lambda u) = 0,\lim\limits_{\lambda\rightarrow +\infty}J(\lambda u) = -\infty$;

$\mathrm{(2)}$ There exists a unique $\lambda_*>0$ such that ${\left. {\frac{d}{{d\lambda }}J(\lambda u)} \right|_{\lambda = {\lambda ^*}}} = 0$, and $J\left( \lambda u \right)$ is increasing on $\lambda\in(0,\lambda_*)$, decreasing on $\lambda\in(\lambda_*,+\infty)$;

$\mathrm{(3)}$ $I(\lambda u) \begin{cases} >0,\lambda\in(0,\lambda_*)\\ <0,\lambda\in(\lambda_*,+\infty) \end{cases}$ and $I(\lambda_* u) = 0$.

Proof. The proof of this lemma can refer to [24]. Here, we omit it.

From Lemma 2.5, it is easy to see that Nehari manifold is not empty and the definition of $d$ is meaningful.

Lemma 2.6. Assume (2) holds. Let $u\in \mathrm{H}\setminus\{0\}$, $\alpha \in \left( {0,{\alpha ^*}} \right)$ and

then we have

$\mathrm{(i)}$ if $0 < {\left\| {\Delta u} \right\|_2} \le r\left( \alpha \right)$, then $I\left( u \right) > 0$.

$\mathrm{(ii)}$ if $I\left( u \right) \le 0$, then ${\left\| {\Delta u} \right\|_2} > r\left( \alpha \right)$.

Proof. For any constant $y>0$, we have $\log y < y$. Combined with Sobolev embedding inequality, through a direct calculation, we obtain

By the above inequality, it is easy to know that $\mathrm{(i)}$ and $\mathrm{(ii)}$ hold.

Lemma 2.7. Combined with the notation in Lamma 2.6, we have

where, $|\Omega|$ is the measure of $\Omega$, and $B = C_p$ is the optimal embedding constant.

Proof. Obviously, if $r_*$ exists, we have $r_*>0$. So we just have to prove $r(\alpha)<\rho(\alpha)$, $r_*$ exists and $r_*<\infty$, where

For any $u\in \mathrm{H}\setminus\{0\}$, using the Hölder inequality, we have

Noticing that ${C_*} = {C_{p + \alpha }}$, $B = {C_p}$, we get

Hence,

that is, $r\left( \alpha \right) < \rho \left( \alpha \right)$.

Next, we prove $r_*$ exists and $r_*<\infty$, the proof is divided into two cases.

Case1. If $n>4$, then $\alpha \in \left( {0,{\alpha ^*}} \right) = \left( {0,\frac{{2n}}{{n - 4}} - p} \right)$. Since $\rho(\alpha)$ is continuous on $\left[ {0,\frac{{2n}}{{n - 4}} - p} \right]$, we have $r_*$ exists and

Case2. If $n\leq4$, then $\alpha \in \left( {0, + \infty } \right)$. We define

thus,

Let

then,

which shows that $g(\alpha)$ is strictly increasing on $\left( {0,k{B^2}{{\left| \Omega \right|}^{1 - \frac{2}{p}}}} \right)$, and strictly decreasing on $\left( {k{B^2}{{\left| \Omega \right|}^{1 - \frac{2}{p}}},\infty } \right)$.

On the one hand, it is easy to see that

On the other hand, we can get that

Given the monotonicity of $g(\alpha)$, it is easy to see that there is a unique ${\alpha _*} \in \left( {k{B^2}{{\left| \Omega \right|}^{1 - \frac{2}{p}}},\infty } \right)$ such that $g\left( {{\alpha _*}} \right) = 0$. Hence, $g\left( \alpha \right) > 0$ for $\alpha \in \left( {0,{\alpha _*}} \right)$, $g\left( \alpha \right) < 0$ for $\alpha \in \left( {{\alpha _*},\infty } \right)$, $h(\alpha)$ attains its maximum at $\alpha = {\alpha _*}$. Therefore,

From Lemma 2.6 and Lemma 2.7, it is not difficult to get the following corollary.

Corollary 1. Assume (2) holds. Let $u\in \mathrm{H}\setminus\{0\}$, we have

$\mathrm{(i)}$ if $0 < {\left\| {\Delta u} \right\|_2} \le {r_*}$, then $I\left( u \right) > 0$;

$\mathrm{(ii)}$ if $I\left( u \right) \le 0$, then ${\left\| {\Delta u} \right\|_2} \ge {r_*}$,

where $r_*$ is defined in Lemma 2.7.

Lemma 2.8. Assume (2) holds, we have

where $r_*$ is defined in Lemma 2.7.

Proof. By the definition of $d$, we know $u\in \mathcal N$, then $I(u) = 0$. Combined with (10) and $\mathrm{(ii)}$ of Corollary 1, we get

thus, (16) holds.

From the previous definition of $E(t)$, we have the following energy equation

Lemma 2.9. If $u_0\in \mathrm H$, $u_1\in L^2(\Omega)$, $p>2$, $E(0)<d$ and $u$ is a weak solution of problem(1) on $[0,T)$, then

$\mathrm{(i)}$ if $I\left( {{u_0}} \right) > 0$, then $u \in W$;

$\mathrm{(ii)}$ if $I\left( {{u_0}} \right) < 0$, then $u \in V$.

Proof. By the definition of $E(t)$, $J(u)$ with (17), we have

$\mathrm {(i)}$ By contradiction, we assume that there exists ${t_0} \in \left[ {0,T} \right)$, such that $u(t)\in W$ on $[0,t_0)$and $u\left( {{t_0}} \right) \notin W$. By the continuity of $J(u)$ and $I(u)$, we have

Obviously, $J\left( {u({t_0})} \right) = d$ is impossible. If $I\left( {u({t_0})} \right) = 0$ holds, by the definition of $d$, then $J\left( {u({t_0})} \right) \geq d$, which is contradictive with (18). Thus, $u\in W$. $\mathrm {(ii)}$ The proof is similar to $\mathrm {(i)}$, which we omit here.

3.   Local existence of weak solution

In this part, we prove the local existence and uniqueness of weak solution. To prove the local existence of weak solution, firstly, we need to introduce the following lemmas.

Lemma 3.1. [22] For any $\varepsilon>0$, there exists a constant $A>0$, such that the function

satisfies

Here, for every $T>0$, we consider the space

endowed with the norm

Lemma 3.2. For every $T>0$, $u \in \mathcal H$ and every initial data $(u_0,u_1)\in \mathrm H\times L^2(\Omega)$, there exists a unique solution $v \in C\left( {\left[ {0,T} \right],{\rm{H}}} \right)\cap C^1\left([0,T], L^2(\Omega\right) \cap {C^2}\left( {\left[ {0,T} \right],{H^{ - {\rm{2}}}}(\Omega )} \right)$ with ${v_t} \in {L^2}\left( {\left[ {0,T} \right],H_0^1(\Omega )} \right)$, which solves the linear problem

Proof. Applying Garlerkin's method, for every $h\geq1$, let ${W_h} = \rm{span}\left\{ {{w_1},{w_2}, \cdots ,{w_h}} \right\}$, where $\{w_j\}$ is the orthonormal complete system of eigenfunctions of $-\Delta$ in $\mathrm H$ such that $\|w_j\|_2 = 1$ for all $j$. According to their multiplicity of

we denote the related eigenvalues repeated by $\lambda_j$. Let

so that $u_0^h \in {W_h}$, $u_1^h \in {W_h}$, $u_0^h \to {u_0}$ in $\mathrm H$ and $u_1^h \to {u_1}$ in $L^2(\Omega)$ as $h \to \infty$. For all $h\geq1$, we seek $h$ functions $\gamma _1^h, \cdots ,\gamma _h^h \in {C^2}\left[ {0,T} \right]$ such that

solves the problem

where $\eta \in {W_h}$ and $t\geq0$. Taking $\eta = {w_j}$ for $j = 1, \cdots ,h$ in (21), we obtain the following Cauchy problem for a linear ordinary differential equation with unknown $\gamma _j^h$:

where ${\psi _j}\left( t \right) = \int_\Omega {{{\left| u \right|}^{p - 2}}u\log {{\left| u \right|}^k}{w_j}dx \in C\left[ {0,T} \right]}$. For all $j$, the above Cauchy problem has a unique local solution $\gamma _j^h \in {C^2}\left[ {0,T} \right]$, which implies a unique ${v_h}$ defined by (20) satisfying (21).

Let $\eta = {\dot v_h(t)}$ in (21), integrating over $\left[ {0,t} \right] \subset \left[ {0,T} \right]$, we get

for every $h\geq1$. We estimate the last term in the right-hand side of (23). Using Hölder's inequality, we have

Using the fact $\left| {{x^{p - 1}}\log x} \right| \le {\left( {e\left( {p - 1} \right)} \right)^{ - 1}}$ for $0 < x < 1$, while ${x^{ - \mu }}\log x \le {\left( {e\mu } \right)^{ - 1}}$ for $x \geq 1,\; \mu>0$. Choosing $\mu>0$ such that $\frac{{p(p - 1 + \mu )}}{{p - 1}} <\frac{2n}{n-2}< {2^*} = \frac{{2n}}{{n - 4}}$, by a direct calculation and Sobolev inequality, we have

here, it is needs to be noted that $C$ in the text is a general constant, and the $C$ in each row and even in the same row is differemt.

By the Sobolev embedding theorem, we have ${\left\| {{{\dot v}_h}} \right\|_p} \le C{\left\| {\nabla {{\dot v}_h}} \right\|_2}$. Combined with Young's inequality, (24) yields

Recalling the convergence of $u_0^h$ and $u_1^h$, by (23) and (26), we obtain

for every $h\geq1$, where $C$ is independent of $h$. By this uniform estimate, we have

$\left\{ {{v_h}} \right\}$ is bounded in ${L^\infty }\left( {\left[ {0,T} \right],\mathrm H} \right)$;

$\left\{ {{{\dot v}_h}} \right\}$ is bounded in ${L^\infty }\left( {\left[ {0,T} \right],{L^2}\left( \Omega \right)} \right) \cap {L^2}\left( {\left[ {0,T} \right],H_0^1\left( \Omega \right)} \right)$;

$\left\{ {{{\ddot v}_h}} \right\}$ is bounded in ${L^2}\left( {\left[ {0,T} \right],{H^{ - 2}(\Omega)}} \right)$.

Thus, up to a subsequence, we could pass to the limit in (21) satisfying above regularity. Then a weak local solution of problem (19) can be obtained.

Uniqueness follows arguing for contradiction, if $v$ and $w$ were two solutions of (19) which have the same initial date, by subtracting the equations and testing with $v_t-w_t$, we could get

which yields $w = v$. The proof of the lemma is complete.

Now, we begin to prove Theorem 2.2.

Proof of Theorem 2.2. For any $u \in {\mathcal H}$, ${u_0} \in \mathrm H$, $u_1 \in L^2(\Omega)$, let

For any $T>0$, we consider

By Lemma 3.2, for any $u \in {{\mathcal U}_T}$ we could define $v = \Phi (u)$, where $v$ is the unique solution of problem (19). We claim that, for a suitable $T>0$, $\Phi$ is a contractive map satisfying $\Phi ({{\mathcal U}_T}) \subseteq {{\mathcal U}_T}$. Given $u \in {{\mathcal U}_T}$, the corresponding solution $v = \Phi(u)$ satisfies the energy identity for all $t\in(0, T]$ as follows

Now we estimate the last term of (28), we get

Substituting (29) into (28), we obtain

Choosing $T>0$ small enough, such that $CT(1 + {R^{2(p - 1 + \mu )}}) \le \frac{{{R^2}}}{2}$. Hence, ${\left\| \Phi(u) \right\|_{\mathcal H}} \le R$, that is $\Phi ({{\mathcal U}_T}) \subseteq {{\mathcal U}_T}$.

Next, we show that $\Phi$ is contractive in ${{\mathcal U}_T}$. Namely, For any ${u_1},{u_2} \in {{\mathcal U}_T}$, there exists $0 < \delta < 1$, such that

Let ${u_1},{u_2} \in {{\mathcal U}_T}$, ${v_1} = \Phi ({u_1})$, ${v_2} = \Phi ({u_2})$ and $z = {v_1} - {v_2}$, then $z$ is the unique solution to the following problem

Multiplying both sides of (30) by $z_t$, and integrating over $(0, t)\times\Omega$, we get

Using Lagrange Theorem, for $0 < \theta < 1$, combined with Lemma 3.1, we have

Since ${u_1},{u_2} \in {{\mathcal U}_T}$, utilizing Hölder's inequality and Sobolev embedding theorem, we have

Choosing $\varepsilon>0$ small enough, such that $\bar p = 2(p - 1) + \frac{{2\varepsilon (p - 1)}}{{p - 2}} < \frac{{2n}}{{n - 4}}$, by a calculation similar to (31), we obtain

From the above calculation, we can deduce

Hence, for some $\delta < 1$ as long as $T$ is sufficiently small, we have

That is,

So by the contraction mapping principle, we can conclude that problem (1) admits a unique solution.

4.   Global existence and energy decay estimate

In this section, we prove Theorem 2.3, which is divided into 4 steps.

Proof of Theorem 2.3. Step 1. Global existence for the case of $E(0)<M$ and $I(u_0)\geq 0$.

By the definition of $E(t)$ and (10), we know that $0\leq J(u_0)\leq E(0)<M$, and combine with Lemma 2.8, then we have respectively

(i) If $E\left( 0 \right) = 0$ and $I\left( {{u_0}} \right) \ge 0$, then this implies that $\left( {{u_0},{u_1}} \right) = \left( {0,0} \right)$, which is a trivial case;

(ii) If $0 < E\left( 0 \right) < M\leq d$ and $I\left( {{u_0}} \right) = 0$, then it contradicts with the definition of potential depth $d$.

Hence, we only need to consider the case of $0 < E\left( 0 \right) < M \leq d$ and $I\left( {{u_0}} \right) > 0$.

Let $\{w_j(x)\}$ be a system of base functions in $\mathrm H$. We construct the following approximate solutions to problem (1)

satisfying

Now, multiplying (33) by $g_{jmt}\left( t \right)$, summing for $j$, and integrating over $\left[ {0,t} \right]$, we can compute

for sufficiently large $m$. Since $E(0)<M\leq d$ and $I(u_0)>0$, by (34) and (35), for sufficiently large $m$, we conclude that $E_m(0) < M\leq d$ and $I(u_{0m})>0$. By the argument in the proof of Lemma 2.9, for sufficiently large $m$ and $0 \leq t<\infty$, we have $u_m(t)\in W$. Hence, combined with (10), we have

for sufficiently large $m$ and $0 \le t < \infty$. So it follows that

By (37), (38), there exist functions $u$ and a subsequence of $\left\{ {{u_m}} \right\}_{m = 1}^\infty$ which we still denote it by $\left\{ {{u_m}} \right\}_{m = 1}^\infty$ such that

By Aubin-Lions-Simon Lemma (see [35], Corollary 4), we get

so, ${u_m} \to u$ a.e. $\left( {x,t} \right) \in \Omega \times \left[ {0,\infty } \right)$, $m \to + \infty$. This implies

On the other hand, from (41), (25) and Lions Lemma (see [30], Lemma 1.3, p.12), we have

Integrating (33) with respect to $t$ we obtain

therefore, up to a subsequence, by (39)-(42), we could pass to the limit in (43). Moreover, from(34) and (35), we get $u(x,0) = u_0$ in $\mathrm H$ and $u_t(x,0) = u_1$ in $L^2(\Omega)$. Then we have a global weak solution $u(x,t)$ to problem(1).

Step 2. Global existence for the case of $E(0) = M$ and $I(u_0)\geq0$.

By Lemma 2.8, we know $d\geq M$. If $E(0) = M<d$, then the problem (1) has a global weak solution, which is similar to the proof of step 1. If $E(0) = M = d$, we consider two cases $I(u_0) = 0$ and $I(u_0)> 0$.

(i) $E(0) = M = d$, $I(u_0) = 0$

From the definition of $d$, we have $J\left( {{u_0}} \right) \ge d$. However, $\frac{1}{2}\left\| {{u_1}} \right\|_2^2 + J\left( {{u_0}} \right) = E\left( 0 \right) = d$, it follows that $J\left( {{u_0}} \right) < d$. So case (1) is impossible.

(ii) $E(0) = M = d$, $I(u_0)>0$

In order to prove the global existence result of problem (1), we first choose a sequence $\left\{ {{\gamma _m}} \right\}_{m = 1}^\infty \subset \left( {0,1} \right)$ such that $\mathop {\lim }\limits_{m \to \infty } {\gamma _m} = 1$. Then we considering the following problem

where ${u_{0m}} = {\gamma _m}{u_0}$, ${u_{1m}} = {\gamma _m}{u_1}$. Since $I\left( {{u_0}} \right) > 0$, it follows from Lemma 2.5 that ${\lambda _*} > 1$.

Hence, we get

and

Using the similar arguments as previous step 1, we find that problem (44) admits a global weak solution $u_m$ which satisfies

and

for any $\upsilon \in\mathrm H$, and for a.e. $0 \le t < \infty$. The remainder of the proof can be processed similarly as previous step 1. Hence, $u$ is a global weak solution for problem (1).

Step 3. Polynomial decay estimate of energy for the case of $E(0)\leq M$ and $I(u_0)\geq0$.

Firstly, by (10), (17) and $I(u)\geq0$, we obtain

A combination of (45) and $E\left( 0 \right) \leq M$, we have

where $\lambda_1$ is the first eigenvalue of the following problem:

for $\phi(x) \in H_0^1\left( \Omega \right)$. Next, multiplying the first equation of problem (1) by $u$ and integrating over $\Omega \times \left( {0,t} \right)$. Using Young inequality, we have

where $C_1$ stand by the best constant in the embedding $\mathrm H\hookrightarrow H_0^1(\Omega)$. Using (45), (46) and $u \in {L^\infty }\left( {0,T;\mathrm H} \right)$, then (47) implies that

From $I\left( u \right) \geq 0$, we know that there exists a ${\lambda _*} \geq1$ such that $I\left( {{\lambda _*}u} \right) = 0$. On the other hand, we have

Hence, we obtain

Combining the above equation with (48), we obtain

and

Differentiating $E\left( t \right)$ and using Eq.(1), we can compute

Since

then integrating (51) over $\left( {0,t} \right)$, we have

Thus, applying $E\left( 0 \right) \leq M$, (46), (48), (49) and (50) to (52), we can derive that there exists a positive constants ${\mathcal K_0}$ such that the energy functional $E\left( t \right)$ satisfies the following polynomial decay estimation:

Step 4. Exponential decay estimate of energy for the case of $E\left( 0 \right) < \min \left\{ {M,\frac{{p - 2}}{{2p}}{{\left( {\frac{{e\mu }}{{k{C_2}^{p + \mu }}}} \right)}^{\frac{2}{{p + \mu - 2}}}}} \right\}$.

We define

for any $0\leq t<\infty$, where $\epsilon$ is a positive constant to be specified later. By the Young inequality, we can easily know that there exist two positive constant ${\alpha _1}$ and ${\alpha _2}$ such that

that is to say, $L\left( t \right)$ and $E\left( t \right)$ are equivalent.

By taking the time derivative of the function $L(t)$, using Eq.(1), we get

By virtue of the Sobolev embedding inequality and (45), we obtain

and

where ${C_2}$, ${C_3}$ are the Sobolev constant satisfying ${\left\| u \right\|_{p + \mu }} \le {C_2}{\left\| {\Delta u} \right\|_2}$, ${\left\| u \right\|_p} \le {C_3}{\left\| {\Delta u} \right\|_2}$. Substituting (56) and (57) into (55), we get

Since $E\left( 0 \right) < \frac{{p - 2}}{{2p}}{\left( {\frac{{e\mu }}{{k{C_2}^{p+\mu}}}} \right)^{\frac{2}{{p + \mu - 2}}}}$, we have

Taking $\beta >0$ small sufficiently such that

Now, choosing $\epsilon >0$ small sufficiently such that

Thus, combining with (54), we have

Integrating (59) over $\left( {0,t} \right)$, we can deduce that there exist ${\mathcal K_1} = \frac{{L\left( 0 \right)}}{{{\alpha _1}}}$ and ${\mathcal K_2} = \frac{{\beta \varepsilon }}{{{\alpha _2}}}$ such that

This completes the proof of Theorem 2.3.

5.   Finite time blow up

In this section, we prove Theorem 2.4, which implies that the solution $u$ of problem (1) blow up in finite time. Firstly, we need to introduce the following lemma.

Lemma 5.1. [23] Let $F(t)$ be a positive $C^2$ function satisfying the inequality

for some $\alpha>0$. If $F(0)>0$ and $F^{'}(0)>0$, then there exists a time $T^*\leq \frac{F(0)}{\alpha F^{'}(0)}$ such that $\lim\limits_{t\rightarrow T^{*-}}F(t) = \infty$.

Now, let us prove the theorem 2.4.

Proof of Theorem 2.4. By contradiction, we suppose that $u$ is global. For any $T>0$, we consider the auxiliary function $F:\left[ {0,T} \right] \to {R^ + }$ defined by

where $b>0$ and $T_0>0$, which will be specified later.

Obviously, $F\left( t \right) > 0$ for any $t \in \left[ {0,T} \right]$. Through a direct calculation, we obtain

Using Schwarz's inequality and Young inequality, we have

Hence,

Let

by the definition of $E(t)$ and (17), we have

From $I(u_0)<0$ and Lemma 2.9, we know $u\in V$, which implies that $I(u)<0$. By Lemma 2.5, there exists a ${\lambda _*} \in \left( {0,1} \right)$ such that $I\left( {{\lambda _*}u} \right) = 0$. Hence, we have

Combined with (63), we get

Choosing $b>0$ sufficiently small such that $0<b\leq 2d-2E(0)$, we have $\xi(t)\geq 0$.

Thus, by the above discussion, we obtain

By the definition of $F(t)$, $F(0) = \|u_0\|^2_2+T\|\nabla u_0\|_2^2+bT_0^2>0$, we choose $T_0$ sufficiently large, which satisfies

thus, $F'\left( 0 \right) = 2b{T_0} + 2\int_\Omega {{u_0}{u_1}dx \ge 2b{T_0} - \left\| {{u_0}} \right\|_2^2} - \left\| {{u_1}} \right\|_2^2 > 0$.

According to Lemma 5.1, we conclude that

for

Hence, we deduce that

By (60), (66) and (67), we have

which contradicts the assumption of $u$ being global. Hence, the solution $u$ of problem (1) blow up in finite time. This completes the proof of Theorem 2.4.

Acknowledgments

We are very grateful to the anonymous referees for their valuable suggestions that improved the article. The work is supported by the fund of the "Thirteen Five" Scientific and Technological Research Planning Project of the Department of Education of Jilin Province in China [number JJKH20190547KJ and JJKH20200727KJ].