1 Introduction

The main purpose of this paper is to study the existence and blow-up of solutions for the following pseudo-parabolic p(x)-Laplacian equation with logarithmic nonlinearity:

$$\begin{aligned} \left\{ \begin{array}{ll} u_{t}-\Delta u_{t}-\Delta _{p(x)} u=\vert u\vert ^{\left. q{(x}\right) -2} u \log (\vert u\vert ), &{} x \in \Omega , t>0, \\ u(x, t)=0, &{} x \in \partial \Omega , t>0, \\ \left. u(x,0)=u_{0} (x\right) , &{} x \in \Omega , \end{array}\right. \end{aligned}$$
(1)

where \(\Delta _{p(x)}u = \text {div}(\vert \nabla u\vert ^{p(x)-2}\nabla u)\) is the p(x)-Laplacian, \(\Omega \subset {\mathbb {R}}^{N}(N \geq 1)\) is a bounded domain with smooth boundary \(\partial \Omega , u_{0}: \Omega \rightarrow {\mathbb {R}}\) is the initial function, and \(p, q: {\overline{\Omega }} \rightarrow {\mathbb {R}}_+\) are continuous functions which satisfy the following conditions:

$$\begin{aligned} 2<p_{-}< p(x)<p_{+}<q_{-}<q(x)<q_{+}< p^{*}(x),\forall x \in {\overline{\Omega }} \end{aligned}$$

with

$$\begin{aligned} p^*(x):= {\left\{ \begin{array}{ll} +\infty &{} \hbox { if N}\ \leq p_+,\\ \frac{Np(x)}{N-p(x)}&{}\hbox { if N}\ > p_+.\\ \end{array}\right. } \end{aligned}$$

Pseudo-parabolic equations are characterized by the occurrence of a time derivative appearing in the highest order term, which can be wildly used in some physical and biological scenarios, such as the seepage of homogeneous fluids through a fissured rock, the heat conduction involving two temperature systems, the unidirectional propagation of nonlinear, dispersive, long waves, fluid flow in fissured porous media, two phase flow in porous media with dynamical capillary pressure and the aggregation of populations (see [1, 2]). For problem (1), we can also give an example about the non-stationary process in semiconductors in the presence of sources, where \(\Delta u\) represents the linear dissipation of free charge current, \(\Delta u_t-u_t\) represents the free electron density rate and the nonlinear term including the p(x)-Laplacian and logarithmic nonlinearity stands for the source of free electron current (see [3]). In particular, the pseudo-parabolic equations involving the p(x)-Laplacian can be used to study electrorheological fluids which are characterized by their ability to change the mechanical properties under the influence of the external electromagnetic, more introductions on physical motivations can be found in [4, 5] and the references contained therein.

In the past years, many authors make efforts to the investigation of the existence and blow-up of solutions for such kinds of equations. In Chen et al. [6], investigated the following semilinear heat equation with logarithmic nonlinearity

$$\begin{aligned} u_{t}-\Delta u=u\log (\vert u\vert ), \end{aligned}$$
(2)

they obtained the existence and blow-up at \(+\infty\) of solutions of problem (2), they further proved that the global weak solution decayed exponentially in the case of \(p=2\), and decayed algebraically in the case of \(p>2\). Subsequently, Peng and Zhou [7] considered the following initial boundary value problem of semilinear heat equation with logarithmic nonlinearity

$$\begin{aligned} \frac{\partial u}{\partial t}-\Delta u=\vert u\vert ^{p-2}u\log \vert u\vert , \end{aligned}$$
(3)

where p satisfies

$$\begin{aligned} 2<p<2^*:= {\left\{ \begin{array}{ll} +\infty &{} \text {if }n\leq 2,\\ \frac{2n}{n-2}&{}\text {if } n\geq 3, \end{array}\right. } \end{aligned}$$

they obtained the existence of global solution, finite time blow-up and the upper bound of blow-up time of problem (3). For problem

$$\begin{aligned} u_{t}-\Delta u_{t}-\Delta _{p} u=\varphi _{p}(u)\log (\vert u\vert ), \end{aligned}$$
(4)

where \(\varphi _{p}(z)=\vert z\vert ^{p-2}z\). In case of \(2<p<2^*\), the existence or nonexistence of global weak solutions as well as finite time blow-up phenomenon were obtained in [8]. In He et al. [9], considered the following nonlinear equation

$$\begin{aligned} u_{t}-\Delta u_{t}-\Delta _{p} u=\vert u\vert ^{q-2} u \log (\vert u\vert ), \end{aligned}$$
(5)

they obtained result of decay estimation and finite blow-up of solutions where \(2<p<q<p(1+\frac{2}{n})\). Recently, Boudjeriou [10] considered the following nonlinear equation

$$\begin{aligned} u_{t}-\Delta _{p(x)} u=\vert u\vert ^{s(x)-2} u\log (\vert u\vert ), \end{aligned}$$
(6)

which involved the p(x)-Laplacian operator with nonlinearities of variable exponent type. They proved that the local solutions of problem (6) blew up in finite time under suitable conditions. More results about partial differential equations involving the p(x)-Laplacian operator, we refer to [11,12,13,14] and the references therein.

A powerful technique for solving the existence of the above problem is the so-called potential well method, which was established by Tsutsumi [15], Levine [16], Payne and Sattinger in [17]. Liu et al. [18, 19] generalized and improved the method by introducing a family of potential wells which include the known potential well as a special case. On the other hand, the physical interpretation of blow-up phenomena is generally thought of as a dramatic increase in temperature which leads to ignition of a chemical reaction. Many researchers applied different methods to derive the sufficient conditions for finite time and infinite time blow-up result. In particular, Levine [16] studied the abstract equation

$$\begin{aligned} p\frac{du}{dt}=-A(t)u+f(u(t)),t\in [0,+\infty ), \end{aligned}$$

where p and A are positive linear operators defined on a dense subdomain D of a real or complex Hilbert Space, in which they obtained the blow-up solutions, under abstract conditions

$$\begin{aligned} 2(\alpha +1)F(x)\le (x,f(x)),F(u_0(x))>\frac{1}{2}(u_0(x),Au_0(x)) \end{aligned}$$

for every \(x \in D\), where \(F(x)=\int _0^1(f(\rho x),x)d\rho\). This work has been recognized as a creative and elegant tool for giving criteria for the blow-up, which is called the concavity method. Nowadays, it is one of the most useful method for blow-up of solutions for evolution equations.

Inspired by the above works, by using the potential well method and concavity method, we consider the existence and blow-up of solutions for problem (1) which combine with pseudo-parabolic, variable exponent and logarithmic nonlinearity term. To our best knowledge, it is the first attempt to study the properties of the solutions for such kind of equations.

Here we give some important definitions as follows: for \(u_{0} \in W_{0}^{1, p(x)}(\Omega )\), we define the energy functional E and Pohozaev functional J as

$$\begin{aligned} E(u)=\int _{\Omega } \frac{1}{p(x)}\vert \nabla u\vert ^{p(x)} d x+\int _{\Omega } \frac{1}{q^{2}(x)}\vert u\vert ^{q(x)} d x- \int _{\Omega } \frac{1}{q{(x)}}\vert u\vert ^{q(x)} \log (\vert u\vert ) dx, \end{aligned}$$
(7)

and

$$\begin{aligned} J(u)=\int _{\Omega } \frac{1}{p^{*}(x)}\vert \nabla u\vert ^{p(x)} d x-\int _{\Omega } \frac{1}{q(x)}\vert u\vert ^{q(x)} \log (\vert u\vert ) dx. \end{aligned}$$
(8)

Let \({\mathcal {P}}=\left\{ u \in W_{0}^{1, p(x)}(\Omega )\backslash \{0\}: J(u)=0\right\}\) be the Pohozaev manifold [10]. Related to the above manifold we have the real number \(\Gamma ={\text {inf }}\{ E(u):u \in {\mathcal {P}}\} .\)

To introduce our main results, we first give the definition of weak solutions and blow-up for problem (1).

Definition 1

(Weak solution) \(u=u(x,t)\in L^{\infty }(0,T; W_{0}^{1, p(x)}(\Omega ))\) with \(u_{t} \in L^{2}\left( 0, T; L^{2}(\Omega )\right)\), is said to be a weak solution of problem (1) on \(\Omega \times [0,T)\), if it satisfies the initial condition \(u(x, 0)=u_{0}(x)\), and the following equality holds

$$\begin{aligned} \int _{\Omega } u_{t} w d x+\int _{\Omega } \nabla u_{t} \nabla w d x+\int _{\Omega }\vert \nabla u\vert ^{p(x)-2}\nabla u \nabla w d x =\int _{\Omega }\vert u\vert ^{q(x)-2} u \log (\vert u\vert )wdx, \end{aligned}$$
(9)

for all \(\omega \in W_{0}^{1, p(x)}(\Omega )\), and for almost every \(t \in (0,T)\).

Definition 2

(Finite time blow-up) Let u(x, t) be a weak solution of problem (1). We say u(x, t) blows up in finite time if the maximal existence time T is finite and \(\lim _{t \rightarrow T}\Vert u\Vert _{H_{0}^{1}(\Omega )}^{2}=+\infty\).

2 Preliminaries and Lemmas

In this section, we will recall some important results of variable exponentials on Lebesgue or Sobolev spaces. For more details on variable exponential Sobolev spaces, please refer to [20,21,22,23].

Let \(\Omega \subset {\mathbb {R}}^{N}\) be a smooth bounded domain and \(p \in L^{\infty }(\Omega )\) be a measurable function with \(p_{+}=ess\sup _{x \in \Omega } p(x)\) and \(p_{-}=ess\inf _{x \in \Omega } p(x)\).

The variable exponent Lebesgue space \(L^{p(x)}(\Omega )\) is defined by

$$\begin{aligned} L^{p(x)}(\Omega )=\left\{ u(x) \mid u \ \text {is measurable in} \Omega \text {and} \int _{\Omega }\vert u\vert ^{p(x)} d x<\infty \right\} \end{aligned}$$

with the norm

$$\begin{aligned} \Vert u\Vert {_{L^{p(x)}}}_{(\Omega )}=\inf \left\{ \lambda >0, \int _{\Omega }\left| \frac{u}{\lambda }\right| ^{p{(x)}} d x \leq +\infty \right\} . \end{aligned}$$

The variable exponent Sobolev space \(W^{1,p(x)}(\Omega )\) is defined by

$$\begin{aligned} W^{1, p{(x)}}(\Omega )=\left\{ u(x) \in L^{p(x)}(\Omega )\big \vert \vert \nabla u \mid \in L^{{p(x)}}(\Omega )\right\} \end{aligned}$$

with the norm

$$\begin{aligned} \Vert u\Vert _{ W^{1,p{(x)}}(\Omega )}=\Vert u\Vert _{L^{p (x)}(\Omega )}+\Vert \nabla u\Vert _{L^{p(x)}(\Omega )}. \end{aligned}$$

So with these norms, the space \(L^{p(x)}(\Omega )\) and \(W^{1,p(x)}(\Omega )\) are reflexive and separable Banach spaces. The closure of \(C_0^{\infty }(\Omega )\) in \(W^{1,p(x)}(\Omega )\) is denoted by \(W_0^{1,p(x)}(\Omega )\), this reflexive Banach space is equipped with the norm \(\vert \vert u\vert \vert _{W_0^{1,p(x)}(\Omega )}=\vert \vert \nabla u\vert \vert _{ L^{p(x)}(\Omega )}\).

By virtue of the notion \(p_+=ess\sup _{x\in \Omega }p(x)\) , \(p_-=ess\inf _{x\in \Omega }p(x)\) and \(\left. 1 \leq p_{-} \leq p(x\right) \leq p_{+}\) a.e. in \(x \in \Omega\), we have

$$\begin{aligned} \Vert u\Vert _{W_{0}^{1,p(x)}(\Omega )}^{p_{-}} \leq \int _{\Omega }\vert \nabla u( x)\vert ^{p (x)} d x\leq \Vert u\Vert _{W_{0}^{1,p(x)} (\Omega )}^{p_{+}},\text {if} \vert \vert u\vert \vert _{W_0^{1,p(x)}(\Omega )}<1, \\ \Vert u\Vert _{W_{0}^{1,p(x)}(\Omega )}^{p_{+}} \leq \int _{\Omega }\vert \nabla u( x)\vert ^{p (x)} d x\leq \Vert u\Vert _{W_{0}^{1,p(x)}(\Omega )}^{p_{-}},\text {if} \vert \vert u\vert \vert _{W_0^{1,p(x)}(\Omega )}\geq 1. \end{aligned}$$

The dual space of \(W^{1,p(x)}(\Omega )\) is denoted by \(W^{-1,p^{\prime }(x)}(\Omega )\), where \(\frac{1}{p(x)}+\frac{1}{p^{\prime }(x)}=1, \forall x \in {\overline{\Omega }}.\)

Related to the properties of logarithms and Lebesgue space \(L^{p(x)}(\Omega )\) , we have the following logarithm inequality and generalized Hölder-type inequality.

Lemma 1

(Logarithm inequality [10]) The following inequality holds

$$\begin{aligned} \vert \log (s)\vert \leq \frac{1}{q(x)} s^{q(x)}, \end{aligned}$$

for all \(s \in [1,+\infty )\) and \(q:{\overline{\Omega }} \rightarrow (0,+\infty )\) is a continuous function satisfy that

$$\begin{aligned} 0<q^{-}=\min _{x \in {{\overline{\Omega }}}} q(x) \leq q(x) \leq q^{+}=\max _{x \in {{\overline{\Omega }}}} q(x)<+\infty . \end{aligned}$$

Lemma 2

(Hölder inequality [20]) For any \(u \in L^{p(x)}(\Omega )\) and \(\nu \in L^{p^{\prime }(x)}(\Omega )\), then

$$\begin{aligned} \left| \int _{\Omega } u \nu d x\right| \leq \left( \frac{1}{p_{-}}+\frac{1}{p^{\prime }_{-}}\right) \Vert u\Vert _{p (x)}\Vert v\Vert _{p^{\prime }(x)}. \end{aligned}$$

We conclude this section by recalling the classical interpolation inequality.

Lemma 3

(Interpolation inequality [24]) If \(1 \leq p_{0}<p_{\theta }<p_{1} \leq \infty\), then

$$\begin{aligned} \Vert u\Vert _{p_{\theta }} \leq \Vert u\Vert _{p_{0}}^{1-\theta }\Vert u\Vert _{p_{1}}^{\theta } \end{aligned}$$

for all \(u \in L^{p_{0}}(\Omega ) \cap L^{p_{1}}(\Omega )\) with \(\theta \in (0,1)\) defined by \(\frac{1}{p_{\theta }}=\frac{1-\theta }{p_{0}}+\frac{\theta }{p_{1}}.\)

3 Main Results

In this section, we will give the three theorems which are, respectively, related to global existence, local existence and finite time blow-up of solutions involving problem (1). The existence or nonexistence of the global solution is obtained by using the potential well method and the logarithmic inequality. The finite time blow-up phenomenon is obtained by concavity method.

Theorem 1

(Global existence) Let \(u_{0} \in W_{0}^{1, p(x)}(\Omega )\) and \(E\left( u_{0}\right) <\Gamma , J\left( u_{0}\right) >0\), then the problem (1) has a global weak solution u(x, t) on \(\Omega \times [0,+\infty )\).

Proof

First we define the potential well associated with the problem (1) as

$$\begin{aligned} W=\{u \in W_{0}^{1,p(x)}(\Omega ):E(u)<\Gamma , J(u)>0\} \cup \{0\}. \end{aligned}$$
(10)

In the space \(W_{0}^{1, p(x)}(\Omega )\), we take a basis \(\left\{ w_{j}\right\} _{j=1}^{\infty }\) and define the finite dimensional space

$$\begin{aligned} V_{m}={\text {Span}}\left\{ w_{1}, w_{2}, \cdots , w_{m}\right\} . \end{aligned}$$
(11)

Let \(u_{0m} \in V_{m}\), then \(u_{0m}=\sum _{j=1}^{m} a_{m i} w_{i} \rightarrow u_{0}\) strongly in \(W_{0}^{1, p(x)}(\Omega )\) as \(m \rightarrow +\infty\). We look for the approximate solutions of the following form

$$\begin{aligned} u_{m}(x, t)=\sum _{j={1}}^{m} \alpha _{mj}(t) w_{j}(x), m=1,2, \cdots , \end{aligned}$$
(12)

where the coefficients \(\alpha _{m j}(t)=\left( u_{m}, w_{j}\right) _{2}\) satisfy the system of ODES

$$\begin{aligned} \begin{aligned}&\quad \int _{\Omega } u_{mt} w d x+\int _{\Omega } \nabla u_{mt} \nabla w d x +\int _{\Omega }\vert \nabla u_m\vert ^{p(x)-2}\nabla u_m \nabla w d x\\&=\int _{\Omega }\vert u_m\vert ^{q(x)-2} u_m \log (\vert u_m\vert )wdx, \end{aligned} \end{aligned}$$
(13)
$$\begin{aligned} \alpha _{m j}(0)=a_{mj}, j=1,2,\ldots , m. \end{aligned}$$
(14)

The existence of local solution of system (13, 14) is guaranteed by Peano’s theorem. Multiplying the equality of (13) by \(\alpha ^{\prime }_{m_{j}}(t)\) , summing for j from 1 to m and integrating with respect to time from 0 to t, it yields

$$\begin{aligned} \int _{0}^{t}\left\| u_{s m}(s)\right\| _{H_{0}^1(\Omega )}^{2} d s+E\left( u_{m}(t)\right) =E\left( u_{0 m}\right) , t \in \left[ 0, t_{0, m}\right] . \end{aligned}$$
(15)

From (12, 13, 15) and the continuity of E, we get \(E(u_{m}(0)) \rightarrow E(u_{0})\). According to the assumption that \(E\left( u_{0}\right) <\Gamma\), we have \(E\left( u_{0m}\right) <\Gamma\) for sufficiently large m. Therefore, we obtain

$$\begin{aligned} \int _{0}^{t}\left\| u_{s m}(s)\right\| _{H_{0}^1(\Omega )}^{2} d s+E\left( u_{m}(t)\right) <\Gamma , t \in \left[ 0, t_{0, m}\right] , \end{aligned}$$
(16)

for sufficiently large m. We will show that \(t_{0,m}=+\infty\) and

$$\begin{aligned} u_m{(t)}\in W, \forall t\geq 0, \end{aligned}$$
(17)

for sufficiently large m. Suppose that (17) does not hold and let \(t_{*} \in \left[ 0, t_{0m}\right]\) be the smallest time for which \(u_m{(t_{*})}\notin W,\) then by the continuity of the \(u_m{(t)}\), we get \(u_{m}( t_{*}) \in \partial W\). Hence, it follows that

$$\begin{aligned} E\left( u_{m}\left( t_{*}\right) \right) =\Gamma \end{aligned}$$
(18)

or

$$\begin{aligned} J\left( u_{m}(t_*)\right) =0. \end{aligned}$$
(19)

If (18) is true, it contradicts with (16). While if (19) is true, then \(u_{m}( t_{*}) \in {\mathcal {P}}\), \( E ({u_m}(t_*)) \geq \inf _{u \in {\mathcal {P}}} E(u)=\Gamma\), which also contradicts with (16). Consequently, we have \(u_m{(t)}\in W, \forall t\geq 0.\)

On the other hand, since \(u_m{(t)}\in W\) and (7), it follows that

$$\begin{aligned} \begin{aligned} E(u_m)&\geq \int _{\Omega } \frac{1}{p(x)}\vert \nabla u_m\vert ^{p(x)} d x- \int _{\Omega } \frac{1}{q{(x)}}\vert u_m\vert ^{q(x)} \log (\vert u_m\vert )dx\\&\geq \int _{\Omega } \frac{1}{p_+}\vert \nabla u_m\vert ^{p(x)} d x- \int _{\Omega } \frac{1}{p^*{(x)}}\vert \nabla u_m \vert ^{p(x)} dx\\&\geq \frac{p_{-}^{*}-p_{+}}{p_{-}^{*}p_{+}} \int _{\Omega }\left| \nabla u_m\right| ^{p(x)}dx. \end{aligned} \end{aligned}$$

Combining the above inequality with (16), we derive

$$\begin{aligned} \int _{0}^{t}\left\| u_{{\text {sm}}}(m)\right\| _{H_0^1(\Omega )}^{2} d s<\Gamma \ \ \ \text {and} \ \ \ \int _{\Omega }\vert \nabla u_m\vert ^{p(x)} d x<\frac{p_{+}p_-^{*}\Gamma }{p_{-}^{*}-p_{+}}, \end{aligned}$$
(20)

from where it follows that \(t_{0,m}=+\infty .\)

By (20), there exists functions u, \(\chi\) and a subsequence of \(\left\{ u_{m}\right\} _{m=1}^{\infty }\) which we still denote by \(\left\{ u_{m}\right\} _{m=1}^{\infty }\) such that

$$\begin{aligned} u_{m} \rightarrow u \text {\ weakly star in} \ L^{\infty }\Big ((0,+\infty );W_{0}^{1, p(x)}(\Omega )\Big ), \end{aligned}$$
(21)
$$\begin{aligned} u_{mt} \rightarrow u_{t} \text { weakly star in} \ L^{2}\Big ((0,+\infty );H_{0}^{1}(\Omega )\Big ), \end{aligned}$$
(22)
$$\begin{aligned} -\text {div}(\vert \nabla u_{m}\vert ^{p(x)-2}\nabla u_{m}) \rightarrow \chi (t)\ \text { weakly star in } L^{\infty }\Big ((0,+\infty );W_{0}^{-1, p^{\prime }(x)}(\Omega )\Big ). \end{aligned}$$
(23)

By (21, 22) and Aubin–Lions–Simon Lemma ([25], Corollary 4), we get

$$\begin{aligned} u_{m} \rightarrow u\ \text { strong in} \ \left. C\Big ((0,+\infty ), L^{r(x)}(\Omega )\right) , \quad \forall r(x) \, \in \left[ 2, p^{*}(x)\right) , x \in {\overline{\Omega }}, \end{aligned}$$
(24)

thus

$$\begin{aligned} u_{m} \rightarrow u \text{ a.e. } x \in \Omega , \forall t>0. \end{aligned}$$
(25)

This implies

$$\begin{aligned} \vert u_{m}\vert ^{q(x)-2}u_m\log (\vert u_m\vert ) \rightarrow \vert u\vert ^{q(x)-2}u\log (\vert u\vert ) \text{ a.e. } x \in \Omega , \forall t>0. \end{aligned}$$
(26)

On the other hand, by a direct calculation, we have

$$\begin{aligned} \begin{aligned}&\quad \int _{\Omega } \Big \vert \vert u_m\vert ^{q(x)} \log (\vert u_m\vert )\Big \vert ^\frac{q_+}{q_+-1}dx\\&\leq \int _{x\in \Omega ,\vert u_m\vert \ge 1} \Big \vert \vert u_m\vert ^{q(x)-1} \log (\vert u_m\vert )\Big \vert ^\frac{q_+}{q_+-1}dx\\&\quad +\int _{x\in \Omega ,\vert u_m\vert \le 1} \Big \vert \vert u_m\vert ^{q(x)-1} \log (\vert u_m\vert )\Big \vert ^\frac{q_+}{q_+-1}dx\\&\leq \int _{x\in \Omega ,\vert u_m\vert \ge 1} \Big \vert \vert u_m\vert ^{q(x)-1} \log (\vert u_m\vert )\Big \vert ^\frac{q_+}{q_+-1}dx+c\vert \Omega \vert \\&\leq \int _{x\in \Omega ,\vert u_m\vert \ge 1} \Big \vert \vert u_m\vert ^{q(x)} \Big \vert ^\frac{q_+}{q_+-1}dx+c\vert \Omega \vert \\&\leq c\vert \Omega \vert +c\vert \Omega \vert \cdot \vert \vert u_m\vert \vert _{L^{p^*}(\Omega )}\\&\leq c. \end{aligned} \end{aligned}$$

Hence, by Lion’s Lemma (see [25], Lemma 1.3), it yields

$$\begin{aligned} \left| u_{m}\right| ^{q(x)-2} u_{m} \log \left( \left| u_{m}\right| \right) \longrightarrow \vert u\vert ^{q(x)-2} u \log (\vert u_m\vert ), \end{aligned}$$
(27)

weakly star in \(L^{\infty }\left( (0,+\infty ); \,L^{q^{\prime }(x)}(\Omega )\right) .\)

By (21, 22, 23) and (27), passing to the limit in (10) and (12) as \(m\rightarrow \infty\), we can show that u satisfies the initial condition \(u(0)=u_{0}\) and

$$\begin{aligned} \int _{\Omega } u_{t} w d x+\int _{\Omega } \nabla u_{t} \nabla w d x+<\chi {(t)},w>=\int _{\Omega }\vert u\vert ^{q(x)-2} u\log (\vert u \vert ) w d x. \end{aligned}$$
(28)

The next step is to prove that

$$\begin{aligned} \chi (t)=-\text {div}(\vert \nabla u\vert ^{p(x)-2} \nabla u). \end{aligned}$$
(29)

Let \(Au=-\text {div}(\vert \nabla u\vert ^{p(x)-2} \nabla u)\), we observe \(A:W_0^{1,p(x)}(\Omega ) \rightarrow W_0^{-1,p'(x)}(\Omega )\) is bounded, monotone and hemicontinuous operator (see [21]). In addition,

$$\begin{aligned} \langle Au,u \rangle =\int _{\Omega }\vert \nabla u\vert ^{p(x)}dx,\forall u\in W_0^{1,p(x)}(\Omega ). \end{aligned}$$

From (27), we have

$$\begin{aligned} \int _0^T\int _{\Omega }\left| u_{m}\right| ^{q(x)-2} u_{m} \log \left( \left| u_{m}\right| \right) \omega dxdt \rightarrow \int _0^T\int _{\Omega }\vert u\vert ^{q(x)-2} u \log (\vert u\vert )\omega dxdt, \end{aligned}$$
(30)

for any \({\omega }\in L^{q_+}(\Omega )\). We observe that

$$\begin{aligned} \begin{aligned}&\quad \Big \vert \int _{\Omega }\vert u_m\vert ^{q(x)}\log (\vert u_m\vert ) -\vert u\vert ^{q(x)}\log (\vert u\vert )dx\Big \vert \\&\le \Big \vert \int _{\Omega }(u_m-u)\vert u_m\vert ^{q(x)-2}u_m\log (\vert u_m\vert ) dx\Big \vert \\&\quad +\Big \vert \int _{\Omega }u\Big (\vert u_m\vert ^{q(x)-2}u_m\log (\vert u_m\vert )-\vert u\vert ^{q(x)-2}u\log (\vert u\vert )\Big )dx\Big \vert . \end{aligned} \end{aligned}$$

From (30), we have

$$\begin{aligned} \Big \vert \int _{\Omega }u\Big (\vert u_m\vert ^{q(x)-2}u_m\log (\vert u_m\vert )-\vert u\vert ^{q(x)-2}u\log (\vert u\vert )\Big )dx\Big \vert \rightarrow 0, \end{aligned}$$
(31)

as \(\ m\rightarrow \infty .\)

Using the Hölder inequality, we get

$$\begin{aligned} \Big \vert \int _{\Omega }(u_m-u)\vert u_m\vert ^{q(x)-2}u_m\log (\vert u_m\vert )dx\Big \vert \le \Vert u_m-u\Vert _{q_+}\Vert \vert u_m\vert ^{q(x)}u_m\log (\vert u_m\vert )\Vert _{\frac{q_+}{q_+-1}}. \end{aligned}$$

From (24) we have

$$\begin{aligned} \Big \vert \int _{\Omega }(u_m-u)\vert u_m\vert ^{q(x)-2}u_m\log (\vert u_m\vert )dx\Big \vert \rightarrow 0, \end{aligned}$$

\(\text {as}\ m\rightarrow \infty .\)

This combined with (31) implies that

$$\begin{aligned} \int _{\Omega }\vert u_m\vert ^{q(x)}\log (\vert u_m\vert )dx \rightarrow \int _{\Omega }\vert u\vert ^{q(x)}\log (\vert u\vert )dx. \end{aligned}$$
(32)

Integrating (28) with respect to time from 0 to T, we can obtain

$$\begin{aligned} \int _0^T\int _{\Omega } u_{t} wdxdt+\int _0^T\int _{\Omega } \nabla u_{t} \nabla wdxdt+\int _0^T \langle \chi (t),w \rangle dt \end{aligned}$$
$$\begin{aligned} =\int _0^T\int _{\Omega }\vert u\vert ^{q(x)-2} u\log (\vert u \vert ) w dxdt. \end{aligned}$$
(33)

Taking \(u=w\) in (33), we obtain

$$\begin{aligned} \begin{aligned}&\quad \int _0^T \langle \chi (t),u \rangle dt \\&=\int _0^T\int _{\Omega }\vert u\vert ^{q(x)} \log (\vert u \vert )dxdt-\frac{1}{2}\vert \vert u(T)\vert \vert _{H_0^1(\Omega )}^2+\frac{1}{2}\vert \vert u(0)\vert \vert _{H_0^1(\Omega )}^2. \end{aligned} \end{aligned}$$
(34)

Multiplying the equality of (13) by \(\alpha _{m_{j}}(t)\) , summing over j form 1 to m and integrating over (0, T), it yields

$$\begin{aligned} \begin{aligned}&\quad \frac{1}{2}\vert \vert u_m(T)\vert \vert _{H_0^1(\Omega )}^2-\frac{1}{2}\vert \vert u_m(0)\vert \vert _{H_0^1(\Omega )}^2+\int _0^T\ \langle Au_m,u_m \rangle dt\\&=\int _0^T\int _{\Omega }\vert u_m\vert ^{q(x)} \log (\vert u_m\vert ) dxdt. \end{aligned} \end{aligned}$$
(35)

From (24) and (32), we deduce that

$$\begin{aligned} \begin{aligned}&\quad \lim _{m\rightarrow \infty }\text {sup}\int _0^T \langle Au_m,u_m \rangle dt\\&=\int _0^T \int _\Omega \vert u\vert ^{q(x)} \log (\vert u \vert ) dxdt-\frac{1}{2}\Vert u(T) \Vert _{H_0^1(\Omega )}^2+\frac{1}{2}\Vert u(0)\Vert _{H_0^1(\Omega )}^2. \end{aligned} \end{aligned}$$
(36)

Combing (34) and (36), we obtain

$$\begin{aligned} \lim _{m\rightarrow \infty }\text {sup}\int _0^T \langle Au_m,u_m \rangle dt=\int _0^T \langle \chi (t),u \rangle dt. \end{aligned}$$
(37)

Finally, by (37) and ([26] Lemma 3.2.2) we deduce that (29) holds, which implies

$$\begin{aligned} \int _{\Omega } u_{t} w d x+\int _{\Omega } \nabla u_{t} \nabla w d x+\int _{\Omega }\vert \nabla u\vert ^{p(x)-2} \nabla u \nabla w d x=\int _{\Omega } \vert \left. u\right| ^{q(x)-2} u \log (\vert u \vert ) w d x \end{aligned}$$
(38)

for all \(w\in W_{0}^{1, p(x)}(\Omega ))\) and a.e. \(\ t\in [0,\infty )\).

In view of Definition 1 and the above discussions, we get the global existence of the solution of problem (1). The proof of Theorem 1 is complete. \(\square\)

Theorem 2

(Local existence) Let \(u_{0} \in W_{0}^{1, p(x)}(\Omega )\) and assume \(2<q_-<q(x)<q_{+}<\frac{N p_{-}}{N-p_{-}}, \forall x \in {\overline{\Omega }}\), then the problem (1) admit a local weak solution. Moreover, u satisfy the energy inequality

$$\begin{aligned} \int _{0}^{t}\left\| u_{s}(s)\right\| _{H_{0}^{1}(x)}^{2} d s+E(u(t)) \leq E\left( u_{0}\right) \, a.e.\ \ t \in \left[ 0, T_{*}\right] \end{aligned}$$
(39)

Proof

Similar to the proof of Theorem 1, by means of Galerkin method, we consider the approximate solution of problem (1) as \(u_{m}(x,t)=\sum _{j=1}^{m} \beta _{m j}(t) w_{j}(x)\), which satisfies the following equations

$$\begin{aligned} \begin{aligned}&\quad \int _{\Omega } u_{mt} w_i d x+\int _{\Omega } \nabla u_{mt} \nabla w_i d x+\int _ {\Omega }\vert \nabla u_m\vert ^{p(x)-2} \nabla u_m \nabla w_i d x\\&=\int _{\Omega } \vert \left. u_m\right| ^{q(x)-2} u_m \log \vert u_m\vert w_i d x, \end{aligned} \end{aligned}$$
(40)
$$\begin{aligned} u_m(0)=u_{0m}. \end{aligned}$$
(41)

Due to \(u_{0} \in W_{0}^{1, p(x)}(\Omega )\), then it exists \(\left\{ b_{m j}, j=1,2, \cdots , m\right\}\) such that

$$\begin{aligned} u_{m}(0)=\sum _{j=1}^{m} b_{m j} w_{j} \rightarrow u_{0}\text { strongly in } W_{0}^{1, p(x)}(\Omega ). \end{aligned}$$
(42)

The existence of local solution of system (40) (41) is guaranteed by Peano’s theorem.

Multiplying the equality of (40) by \(\beta _{m j}(t)\) and summing over j from 1 to m, we obtain

$$\begin{aligned} \frac{1}{2} \frac{d}{d t}\left\| u_{m}\right\| _{\mathrm {H}_{0}^{1}(\Omega )}^{2}+\int _{\Omega }\left| \nabla u_{m}\right| ^{p(x)} d x=\int _{\Omega }\left| u_{m}\right| ^{q(x)} \log \left( \left| u_{m}\right| \right) dx. \end{aligned}$$
(43)

By virtue of Lemma 1, we have

$$\begin{aligned} \begin{aligned}&\quad \int _{\Omega }\left| u_{m}\right| ^{q(x)} \log \left( \left| u_{m}\right| \right) d x\\&\leq \int _{\{x \in \Omega \vert \vert u_m\vert>1\}}\left| u_{m}\right| ^{q(x)} \log \left( \left| u_{m}\right| \right) d x\\&\leq \int _{\left\{ x \in \Omega \vert \vert u_{m}\vert >1\right\} }\left| u_{m}\right| ^{q_+} \log \left( \left| u_{m}\right| \right) d x\\&\leq \frac{1}{\mu } \int _{\Omega }\left| u_{m}\right| ^{q_++\mu } d x =\frac{1}{\mu }\left\| u_{m}(t)\right\| _{q_{_+}+\mu }^{q_{+}+\mu }. \end{aligned} \end{aligned}$$

Choosing \(\mu\) such that \(2<p_{-}<\mu <\frac{Np_{-}}{N-p_{-}}-q_{+}\), we have

$$\begin{aligned} \int _{\Omega }\left| u_{m}\right| ^{q(x)} \log \left( \left| u_{m}\right| \right) d x\leq \vert \vert u_m\vert \vert _{2}^{(1-\theta )(q_++\mu )}\vert \vert u_m\vert \vert _{\frac{Np_-}{N-p_-}}^{\theta (q_++\mu )}, \end{aligned}$$
(44)

where \(\theta \in (0,1)\) and \(\frac{1}{\mu +q_{+}}=\frac{\theta (N-p_-)}{N p_{-}}+\frac{1-\theta }{2}\).

Due to \(L^{p*(x)}(\Omega )\hookrightarrow L^{\frac{N p_-}{N-p_-}}(\Omega )\) and \(W^{1,p(x)}(\Omega )\hookrightarrow L^{p*(x)}(\Omega )\), we have

$$\begin{aligned} \vert \vert u_m\vert \vert _{\frac{Np_-}{N-p_-}}\leq c\vert \vert u_m\vert \vert _{W_{0}^{1,p(x)}(\Omega )}. \end{aligned}$$

Combing the above inequality with (44), we obtain

$$\begin{aligned} \int _{\Omega }\vert u_m\vert ^{q(x)}\log (\vert u_m\vert )\leq c\vert \vert u_m\vert \vert _2^{(1-\theta )(q_+\mu )}\vert \vert u_m\vert \vert _{W_0^{1,p(x)}(\Omega )}^{\theta (q_++\mu )}. \end{aligned}$$

By virtue of \(H_0^1(\Omega )\hookrightarrow L^2(\Omega )\), it yields

$$\begin{aligned} \int _{\Omega }\vert u_m\vert ^{q(x)}\log (\vert u_m\vert )dx\leq c\vert \vert u_m\vert \vert _{H_0^1(\Omega )}^{(1-\theta )(q_+\mu )}\vert \vert u_m\vert \vert _{W_0^{1,p(x)}(\Omega )}^{\theta (q_++\mu )}. \end{aligned}$$
(45)

We discuss the proof into two cases as follows.

Case 1 Suppose \(\Vert u_m\Vert \geq 1\), then

$$\begin{aligned} \int _{\Omega }\vert u_m\vert ^{q(x)}\log (\vert u_m\vert )dx\leq c\vert \vert u_m\vert \vert _{H_0^1(\Omega )}^{(1-\theta )(q_++\mu )}\Big (\int _{\Omega }\vert \nabla u_m\vert ^{p(x)}dx\Big )^{\frac{\theta (q_++\mu )}{ p_-}}. \end{aligned}$$
(46)

Due to \(\frac{1}{\mu +q_{+}}=\frac{\theta (N-p_-)}{N p_{-}}+\frac{1-\theta }{2}\), we get

$$\begin{aligned} {1}=\frac{\theta (N-p_-)(\mu +q_{+})}{N p_{-}}+\frac{(1-\theta )(\mu +q_{+})}{2} >\frac{\theta (N-p_-)(\mu +q_{+})}{N p_{-}}. \end{aligned}$$

Setting \(\frac{N-p_-}{Np_-}<\frac{1}{2}\) , we obtain

$$\begin{aligned} \frac{p_-}{\theta (q_++\mu )}>1. \end{aligned}$$

By Young’s inequality and (46), it follows that

$$\begin{aligned} \int _{\Omega }\vert u_m\vert ^{q(x)}\log (\vert u_m\vert )dx\leq c\Big (\vert \vert u_m\vert \vert _{H_0^1(\Omega )}^{2} \Big )^\frac{(1-\theta )(q_++\mu )p_-}{2p_--2\theta (q_+\mu )}+\varepsilon \int _{\Omega }\vert \nabla u_m\vert ^{p(x)}dx. \end{aligned}$$
(47)

Case 2 Suppose \(\vert \vert u_m\vert \vert \leq 1\), then

$$\begin{aligned} \int _{\Omega }\vert u_m\vert ^{q(x)}\log (\vert u_m\vert )dx\leq c\Big (\vert \vert u_m\vert \vert _{H_0^1(\Omega )}^{2}\Big )^\frac{(1-\theta )(q_++\mu )}{2}. \end{aligned}$$
(48)

Due to \(p_--\theta (q_++\mu )\) and \(\frac{p_-}{p_--\theta (q_++\mu )}>1\), it follows that

$$\begin{aligned} \int _{\Omega }\vert u_m\vert ^{q(x)}\log (\vert u_m\vert )dx\leq c+\Big (\vert \vert u_m\vert \vert _{H_0^1(\Omega )}^{2}\Big )^\frac{(1-\theta )(q_++\mu )p_-}{2p_--2\theta (q_++\mu )}. \end{aligned}$$
(49)

By (47) and (49), we obtain

$$\begin{aligned} \int _{\Omega }\vert u_m\vert ^{q(x)}\log (\vert u_m\vert )dx\leq c\Big (\vert \vert u_m\vert \vert _{H_0^1(\Omega )}^{2} \Big )^\frac{(1-\theta )(q_++\mu )p_-}{2p_--2\theta (q_+\mu )}+\varepsilon \int _{\Omega }\vert \nabla u_m\vert ^{p(x)}dx. \end{aligned}$$

Choosing \(\alpha =\frac{(1-\theta )(q_++\mu )p_-}{2p_--2\theta (q_+\mu )}\) and using \(\frac{1}{\mu +q_{+}}=\frac{\theta (N-p_-)}{N p_{-}}+\frac{1-\theta }{2}\), we derive that

$$\begin{aligned} \alpha -1=\frac{\theta p_-(\mu +q_+)}{N[p_--\theta (q_++\mu )]}>0. \end{aligned}$$
(50)

Setting \(\varepsilon \in (0,1)\), we have

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\Vert u_m \Vert _{H_0^1(\Omega )}^{2}+\int _{\Omega }\Vert \nabla u_m \Vert ^{p(x)}dx\leq c\Big (\Vert u_m \Vert _{H_0^1(\Omega )}\Big )^\alpha +\varepsilon \int _{\Omega }\vert \nabla u_m\vert ^{p(x)}dx. \end{aligned}$$

Let \(S_m(t)=\Vert u_m \Vert _{H_{0}^1(\Omega )}^2\), we obtain

$$\begin{aligned} S_m^{\prime }(t)=c(S_m(t))^\alpha . \end{aligned}$$
(51)

Integrating inequality (51) with respect to time from 0 to t, we get

$$\begin{aligned} (S_m(t))^{1-\alpha }\leq c(1-\alpha )t+(S_m(0))^{1-\alpha }. \end{aligned}$$
(52)

Since \((S_m(t))^{1-\alpha }>0\), it yields \(t\leq \frac{S_m^{1-\alpha }(0)}{c(\alpha -1)}=T_*\) and \(S_m(t)=\vert \vert u_m\vert \vert _{H_0^1(\Omega )}^2<c, \forall\) \(t \in [0,T_*].\)

Multiplying the equality of (40) by \(\beta '_{mj}(t)\) and summing over j from 1 to m, and integrating over (0,t) yields

$$\begin{aligned} \int _0^t \vert \vert u_{ms}(s)\vert \vert _{H_0^1(\Omega )}^2ds+E(u_{m}(t))=E(u_{m}(0)). \end{aligned}$$
(53)

By the continuity of the functional E and (41), there exists a constant \(c>0\) satisfying

$$\begin{aligned} E(u_{0m})\leq c, \end{aligned}$$
(54)

for any positive integer m.

On the other hand,

$$\begin{aligned} \begin{aligned} E(u_{m})&=\int _{\Omega }\frac{1}{p(x)}\vert \nabla u_m\vert ^{p(x)}dx+\int _{\Omega }\frac{1}{q^2(x)}\vert u_m\vert ^{q(x)}dx-\int _{\Omega }\frac{1}{q(x)}\vert u_m\vert \log \vert u_m\vert dx\\&\geq \int _{\Omega }\frac{1}{p(x)}\vert \nabla u_m\vert ^{p(x)}dx-\int _{\Omega }\frac{1}{q(x)}\vert u_m\vert ^{q(x)}\log \vert u_m\vert dx\\&\geq \frac{1}{p_-}\int _{\Omega }\vert \nabla u_m\vert ^{p(x)}dx-\frac{1}{q_+}\int _{\Omega }\vert u_m\vert ^{q(x)}\log \vert u_m\vert dx\\&\geq \frac{1}{p_-}\int _{\Omega }\vert \nabla u_m\vert ^{p(x)}dx-\frac{\varepsilon }{q_+}\int _{\Omega }\vert \nabla u_m\vert ^{p(x)}dx-\frac{c}{q_+}(\vert \vert u_m\vert \vert _{H_0^1(\Omega )}^{2})^{\alpha }\\&=\left( \frac{1}{p_-}-\frac{\varepsilon }{q_+}\right) \int _{\Omega }\vert \nabla u_m\vert ^{p(x)}dx-\frac{c}{q_+}(\vert \vert u_m\vert \vert _{H_0^1(\Omega )}^{2})^{\alpha }, \end{aligned} \end{aligned}$$

thus

$$\begin{aligned} (\frac{1}{p_-}-\frac{\varepsilon }{q_+})\int _{\Omega }\vert \nabla u_m\vert ^{p(x)}dx-\frac{c}{q_+}(\vert \vert u_m\vert \vert _{H_0^1(\Omega )}^{2})^{\alpha }<c. \end{aligned}$$
(55)

By (53, 54, 55) and \(\vert \vert u_m\vert \vert _{H_0^1(\Omega )}^2<c\), there exists functions \(u,\chi\) and a subsequence of \(\left\{ u_{m}\right\} _{m=1}^{\infty }\) which we still denote by \(\left\{ u_{m}\right\} _{m=1}^{\infty }\) such that

$$\begin{aligned} u_{m} \rightarrow u \text {\ weakly \ star\ in} \ L^{\infty }\Big ([0,T_*];W_{0}^{1, p(x)}(\Omega )\Big ), \end{aligned}$$
(56)
$$\begin{aligned} u_{mt} \rightarrow u_{t} \text {\ weakly \ star\ in} \ L^{2}\Big ([0,T_*];H_{0}^{1}(\Omega )\Big ), \end{aligned}$$
(57)
$$\begin{aligned} -\text {div}(\vert \nabla u_{m}\vert ^{P(x)-2}\nabla u_{m}) \rightarrow \chi (t)\ \text {weakly star in} \ L^{\infty }\Big ([0,T_*];W_{0}^{-1, p^{\prime }(x)}(\Omega )\Big ). \end{aligned}$$
(58)

The remaining proof of the existence of local solutions is similar as described in Theorem 1 , then it yields

$$\begin{aligned} \int _{\Omega } u_{t} w d x+\int _{\Omega } \nabla u_{t} \nabla w d x+\int _{\Omega }\vert \nabla u\vert ^{p(x)-2} \nabla u \nabla w d x=\int _{\Omega } \vert \left. u\right| ^{q(x)-2} u \log \vert u\vert w d x, \end{aligned}$$

for all \(\omega \in W_{0}^{1, p(x)}(\Omega )\), a.e. in \([0,T_*]\).

In view of the Definition 1, we get the local existence of the solution of problem (1). Now, let’s prove that the solution satisfies the energy inequality. By (53), we get

$$\begin{aligned} \int _{0}^{T_*}\theta (t)dt\int _0^t \Vert u_{ms}(s)\Vert _{H_0^1(\Omega )}^2ds+\int _{0}^{T_*}E(u_{m}(t))\theta (t)dt=\int _{0}^{T_*} E(u_{m}(0))\theta (t)dt. \end{aligned}$$
(59)

Due to the second term of the left-hand side \(\int _{0}^{T_*}E(u_{m}(t))\theta (t)dt\) is lower semi-continuous with respect to the weak topology of \(W_{0}^{1, p(x)}(\Omega )\), then it follows that

$$\begin{aligned} \int _{0}^{T_*}E(u(t))\theta (t)dt \leq \varliminf _{n \rightarrow \infty } \int _{0}^{T_*}E(u_{m}(t))\theta (t)dt. \end{aligned}$$
(60)

By (59) and (60) and letting \(m\rightarrow +\infty\), we have

$$\begin{aligned} \int _{0}^{T_*}\theta (t)dt\int _0^1 \vert \vert u_{s}(s)\vert \vert _{H_0^1(\Omega )}^2ds+\int _{0}^{T_*}E(u(t))\theta (t)dt\leq \int _{0}^{T_*} E(u(0))\theta (t)dt. \end{aligned}$$

Since \(\theta (t)\) is arbitrary, we get

$$\begin{aligned} \int _{0}^{t}\left\| u_{s}(s)\right\| _{H_{0}^{1}(x)}^{2} d s+E(u(t)) \leq E\left( u_{0}\right) \ , \ \text {a.e.} \ t \in \left[ 0, T_{*}\right] . \end{aligned}$$
(61)

The proof of Theorem 2 is complete. \(\square\)

Theorem 3

(Blow-up) Let \(u_{0} \in W_{0}^{1, p(x)}(\Omega )\) and \(2<q_-< q(x)<q_{+}<\frac{N p_{-}}{N-p_{-}}\). If \(E(u_0)<0,p_+<q(x)=q\), the local weak solution of problem (1) blows up at finite time.

Proof

Defining

$$\begin{aligned} \Gamma (t) =\frac{1}{2}\int _{0}^{t}\vert \vert u(s)\vert \vert ^2_{H_0^1(\Omega )}ds+\frac{T-t}{2}\vert \vert u_0\vert \vert _{H_0^1(\Omega )}^2,0 \leq t <T, \end{aligned}$$
(62)

and then taking the second derivative of \(\Gamma (t)\), we get

$$\begin{aligned} \Gamma^{\prime\prime }(t)=(u_t(t),u(t))+(\nabla u_t(t),\nabla u(t)). \end{aligned}$$

By (38) and taking \(\omega =\mu\), we obtain

$$\begin{aligned} \begin{aligned} \Gamma^{\prime\prime }(t)&=-\int _{\Omega }\vert \nabla u\vert ^{p(x)}dx+\int _{\Omega }\vert u\vert ^{q(x)}\log (\vert u\vert )dx\\&\geq -\int _{\Omega }\vert \nabla u\vert ^{p(x)}dx+\int _{\Omega }\vert u\vert ^{q(x)}\log (\vert u\vert )dx+\int _{\Omega }\left( \frac{q}{p_+}-\frac{q}{p(x)}\right) \vert \nabla u\vert ^{p(x)}dx\\&=\left( \frac{q}{p_+}-1\right) \int _{\Omega }\vert \nabla u\vert ^{p(x)}\vert dx+\frac{1}{q}\int _{\Omega }\vert u\vert ^{q}dx-qE(u(t)). \end{aligned} \end{aligned}$$

It implies \(\Gamma^{\prime\prime }(t)\geq -qE(u(t)).\) Due to the energy equality (61), we have

$$\begin{aligned} \Gamma^{\prime\prime }(t)-\beta \geq -qE((u(t)))+q(E(u_0))\geq q\int _0^t\vert \vert u_s(s)\vert \vert _{H_0^1(\Omega )}^2ds, \end{aligned}$$

where \(\beta =-qE(u_0)\).

By \(\Gamma ^{\prime }(t)=\int _0^t(u_t(s),u(s))_{H_0^1(\Omega )}ds\) and Hölder inequality, we obtain

$$\begin{aligned} \begin{aligned} (\Gamma ^{\prime }(t))^2&\leq \int _0^t\vert \vert u_s(s)\vert \vert _{H_0^1(\Omega )}^2ds\int _0^t\vert \vert u(s)\vert \vert _{H_0^1(\Omega )}^2ds\\&\leq 2\int _0^t\vert \vert u_s(s)\vert \vert _{H_0^1(\Omega )}^2ds\Big (\frac{1}{2}\int _0^t\vert \vert u(s)\vert \vert _{H_0^1(\Omega )}^2ds+\frac{T-t}{2}\vert \vert u_0\vert \vert _{H_0^1(\Omega )}^2\Big )\\&=2\int _0^t\vert \vert u_s(s)\vert \vert _{H_0^1(\Omega )}^2ds(\Gamma (t)),\\ \end{aligned} \end{aligned}$$

then \(\Gamma (t)\Gamma^{\prime\prime }(t)-\frac{q}{2}(\Gamma^{\prime }(t))^2 \geq \Gamma (t)\beta >0, t \in [0,T].\)

Let \(y(t)=(\Gamma (t))^{-\frac{q-2}{2}}\), then

$$\begin{aligned} y'(t)=-\frac{q-2}{2}\Gamma ^{-\frac{q}{2}}(t)\Gamma '(t), \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} y^{\prime }(t)&=-\frac{q-2}{2}\Gamma ^{-\frac{q}{2}}(t)\Gamma^{\prime\prime }(t) +\frac{q}{2}(\frac{q-2}{2})(\Gamma (t))^{-\frac{q+2}{2}}(\Gamma ^{\prime }(t))^{2}\\&=\Gamma^{-\frac{q}{2}}{}(t)(\frac{2-q}{2}\Gamma^{\prime\prime }(t) +\frac{q(q-2)}{4}(\Gamma ^{\prime }(t))^2\Gamma ^{-1}(t))\\&\leq \Gamma ^{-\frac{q}{2}}(t)(\frac{2-q}{2}\Gamma^{\prime\prime }(t) +\frac{(q-2)}{2}(\Gamma^{\prime\prime }(t))-\beta )\\&=\Gamma ^{-\frac{q}{2}}(t)\Big (-\frac{q-2}{2}\beta \Big )<0. \end{aligned} \end{aligned}$$

Since \(y^{\prime \prime }(t)<0, y(t)\) is a concave function and \(y^{\prime }(t)\leq y^{\prime }(0)=0,0\leq t<T.\) And due to \(y^{\prime }(t)<0\) and \(y^{\prime \prime }(t)<0,0\leq t <T,\) it implies as long as T is large enough, there exists \({T_*} \in [0,T)\), \(\lim \limits _{t\rightarrow T_*^-}y(t)=0.\) Consequently, we obtain \(\lim \limits _{t\rightarrow T_*^-}\vert \vert u(s)\vert \vert _{H_0^1(\Omega )}^2=+\infty\).

In view of the Definition 2, we get the blow-up of the solution of problem (1). The proof of Theorem 3 is complete. \(\square\)

4 Conclusion

In this paper, by virtue of the potential well method and logarithmic inequalities, we obtain the global existence of solutions of problem (1). What’s more, the finite time blow-up phenomenon is obtained by concavity method. The first attempt to study the properties of the solutions for such kind of equations will enrich the research of mathematical physics equations.