1 Introduction and the main result
We consider the following semilinear hyperbolic equation with nonstandard growth condition:
where \(\varOmega \subset \mathbb{R}^{N}\) (\(N\geqslant 1\)) is a bounded domain with smooth boundary ∂Ω, \(T>0\). It will be assumed throughout the paper that the exponents \(p(x,t)\), \(m(x,t)\) are continuous in \(Q_{T}=\varOmega \times (0,T)\) and satisfy that
where \(\omega (r)\) satisfies
$$ \textstyle\begin{cases} u_{tt}-\Delta u+\int _{0}^{t}g(t-\tau )\Delta u(\cdot,\tau )\,d\tau + \vert u_{t} \vert ^{m(x,t)-2}u _{t}= \vert u \vert ^{p(x,t)-2}u, &x\in \varOmega , t>0, \\ u(x,t)=0, &x\in \partial \varOmega , t\geq 0, \\ u(x,0)=u_{0}(x),\qquad u_{t}(x,0)=u_{1}(x), &x\in \varOmega , \end{cases} $$
(1)
$$\begin{aligned} &2< p^{-}=\inf_{(x,t)\in Q_{T}}p(x,t)\leqslant p(x,t) \leqslant p^{+}=\sup_{(x,t)\in Q_{T}}p(x,t)< \infty , \end{aligned}$$
(2)
$$\begin{aligned} &2< m^{-}=\inf_{(x,t)\in Q_{T}}m(x,t)\leqslant m(x,t) \leqslant m^{+}=\sup_{(x,t)\in Q_{T}}m(x,t)< \infty , \end{aligned}$$
(3)
$$\begin{aligned} & \bigl\vert p(x,t)-p(y,s) \bigr\vert + \bigl\vert q(x,t)-q(y,s) \bigr\vert \leq \omega \bigl( \vert x-y \vert +\sqrt{ \vert t-s \vert } \bigr), \\ &\quad \forall x,y\in \varOmega ,t,s>0, \vert x-y \vert +\sqrt{ \vert t-s \vert }< 1, \end{aligned}$$
(4)
$$ \limsup_{r\rightarrow 0^{+}}\omega (r)\ln \biggl(\frac{1}{r} \biggr)=C< +\infty . $$
Problem (1) may describe many phenomena of applied science such as electro-rheological fluids, viscoelastic fluids, processes of filtration through a porous media, and fluids with temperature-dependent viscosity; the interested readers may refer to [1, 2, 7, 9, 21] and the references therein. As far as we know, when p and m are fixed constants, many authors discussed the existence, uniqueness, blowing-up, and global existence of solutions to Problem (1). For example, in the absence of the viscoelastic term (\(g=0\)), Georgiev and Todorova in [8] studied the initial boundary value problem
They applied the Galerkin approximation method and the contraction mapping theorem to prove that Problem (5) had a unique global solution for small initial data and \(1< p\leqslant m\), whereas for \(p>m\), they obtained that the solution of Problem (5) blew up in finite time for a negative initial energy by applying energy estimate methods and Gronwall’s inequality. Later, Messaoudi in [16] improved the above results. Roughly speaking, he proved that the solution blew up in finite time for a positive initial energy. However, it is well known that the source term causes finite-time blow-up of solutions and drives the equation to possible instability, while the damping term prevents finite-time blow-up of the solution and drives the equation toward stability. So, it is of interest to explore the mechanism of how the sources dominate the two types of dissipation (the finite-time memory term \(\rm {\int _{0}^{t}g(t-\tau )\Delta u( \cdot ,\tau )\,d\tau }\) and the weak damping term \(|u_{t}|^{m-2}u_{t}\)), which attracts considerable attention. The interaction between the damping term and the source term makes the problem more interesting. In the presence of the viscoelastic term \((g\neq 0)\), Cavalcanti and Soriano [5] obtained a rate of exponential decay to the solution of Problem (1) with the assumption that the kernel g is of exponential decay and \(m=2\) (a localized damping mechanism \(a(x)u_{t}\)). Later, Cavalcanti in [6] and Berrimi and Messaoudi in [3] improved this work by using different methods. In addition, Messaoudi in [18] generalized the results in [3, 5]. For more works, the interested readers may refer to [4, 14‐19] and the references therein. However, there are few results about lower bound for lifespan. Sun, Guo, and Gao in [22] considered some estimates of the lower bound of blow-up time for the following problem:
They applied an energy estimate method and the Sobolev inequalities to give an estimate of the lower bound for the blow-up time when \(2< p\leqslant \frac{2(N-1)}{N-2}\), and later Guo and Liu in [13] obtained an estimate of the lower bound for the blow-up time in the supercritical case \(\frac{2(N-1)}{N-2}< p<\frac{2(N^{2}-2)}{N ^{2}-2N}\). For more works, the interested readers may refer to [23, 24]. When p is a function, the authors in [2, 20] applied Kaplan’s method to establish the nonglobal existence and global existence results for Problem (1) in the absence of the viscoelastic term and the damping term. As far as we know, in the presence of the viscoelastic term (\(g\not \equiv 0\)) and the damping mechanism \(|u_{t}|^{m-2}u_{t}\), the results of blow-up of solutions with positive initial energy are seldom seen for the case with variable exponents. Different from the case with constant exponents, the variable exponent brings us some essential difficulties.
$$ \textstyle\begin{cases} u_{tt}-\Delta u+ \vert u_{t} \vert ^{m-2}u_{t}= \vert u \vert ^{p-2}u, &x\in \varOmega , t>0, \\ u(x,t)=0, &x\in \partial \varOmega , t\geq 0, \\ u(x,0)=u_{0}(x),\qquad u_{t}(x,0)=u_{1}(x), &x\in \varOmega . \end{cases} $$
(5)
$$ \textstyle\begin{cases} u_{tt}-\triangle u -\omega \triangle u_{t}+\mu u_{t}= \vert u \vert ^{p-2}u, &(x,t) \in \varOmega \times [0,T], \\ u(x,t)=0, &(x,t)\in \partial \varOmega \times [0,T], \\ u(x,0)=u_{0}(x),\qquad u_{t}(x,0)=u_{1}(x), &x\in \varOmega . \end{cases} $$
(6)
- How to overcome the lack of the monotonicity of the energy functional constructed in [18] with respect to time variable?
- Owing to the existence of a gap between the norm and the modular (that is, \(\int _{\varOmega }|u|^{p(\cdot)}\,dx\not \equiv \|u\|^{p(\cdot)} _{p(\cdot)}\)), it is not easy to obtain the results similar to those of Lemmas 2.2–2.4 in [18]. In fact, the proof of Theorem 1.2 of [18] depends strongly on the conclusions of Lemmas 2.2–2.4 and the monotonicity of the energy functional. It is unfortunate that we cannot obtain such results in the case with variable exponents.
Anzeige
To bypass the difficulties mentioned above, we have to look for some new methods or techniques to discuss some properties of solutions to the above problem. In this paper, we construct a new control function and apply suitable embedding theorems to prove that the solution blows up in finite time for a positive initial energy. At the same time, we apply the energy estimate method to establish a differential inequality and then obtain an explicit lower bound for blow-up time.
Before stating our main result, we first define some energy functionals. Denote by \(L^{p(\cdot)}(\varOmega )\) the space of measurable functions \(f(x)\) on \(Q_{T}\) such that
The norm of \(f(x)\) in space \(L^{p(\cdot)}(Q_{T})\) is defined as follows:
It is obvious that \(L^{p(\cdot)}(Q_{T})\) is a Banach space [7]. It follows directly from the definition that
By Corollary 3.34 in [7], we have that
where \(1< p^{-}\leq p(\cdot)\leq p^{+}\leq \frac{2N}{N-2}\) (\(N\geq 3\)) and B is the embedding constant.
$$ A_{p(\cdot)}(f)= \iint _{Q_{T}} \bigl\vert f(x) \bigr\vert ^{p(\cdot)}\,dx\,dt< \infty . $$
$$ \Vert f \Vert _{p(\cdot),Q_{T}}\equiv \Vert f \Vert _{L^{p(\cdot)}(Q_{T})}= \inf \biggl\{ \lambda >0 :A _{p(\cdot)}\biggl(\frac{f}{\lambda }\biggr) \leq 1\biggr\} . $$
$$ \min \bigl\{ \Vert f \Vert _{p(\cdot),\varOmega }^{p^{-}}, \Vert f \Vert _{p(\cdot),\varOmega }^{p^{+}}\bigr\} \leq A_{p(\cdot)}(f)\leq \max \bigl\{ \Vert f \Vert _{p(\cdot),\varOmega }^{p^{-}}, \Vert f \Vert _{p(\cdot), \varOmega }^{p^{+}}\bigr\} . $$
(7)
$$ \Vert u \Vert _{p(\cdot)}\leq B \Vert \nabla u \Vert _{2}, $$
(8)
Before proving the main results of this paper, we first state a local existence theorem.
Theorem 1.1
Suppose that the exponents
\(p(x,t)\), \(m(x,t)\)satisfy (2)–(4), and the following conditions hold:
Then Problem (1) has a unique local solution
for someT.
$$\begin{aligned} &(H_{1})\quad \max \bigl\{ m^{+}\cdot p^{+}\bigr\} \leqslant \frac{2(N-1)}{N-2},\quad p_{t} \geqslant 0, \qquad \frac{p_{t}(x,t)}{p^{2}(x,t)}\in L_{\mathrm{loc}}^{1}\bigl((0,\infty );L^{1}(\varOmega )\bigr); \\ &(H_{2})\quad g(t)>0,\qquad g'(t)< 0, \quad t\geqslant 0,\qquad 1- \int _{0}^{\infty }g(s)\,ds=k>0; \\ &(H_{3})\quad (u_{0},u_{1})\in H_{0}^{1}(\varOmega )\times L^{2}(\varOmega ). \end{aligned}$$
$$\begin{aligned}& u\in C\bigl([0,T);H^{1}_{0}(\varOmega )\bigr)\cap L^{p^{-}}\bigl(0,T;L^{p(x,t)}(\varOmega )\bigr),\\& u_{t} \in C\bigl([0,T);L^{2}(\varOmega )\bigr)\cap L^{m}\bigl( \varOmega \times (0,T)\bigr) \end{aligned}$$
Anzeige
The proof of the existence of solutions relays on Galerkin approximation technique and the contraction mapping theorem. For more details, we may refer to [7, 13, 23].
Set
where \(B_{1}=\max \{B, \sqrt{k}\}\).
$$ E_{1}= \biggl(\frac{1}{2}- \frac{1}{p^{-}} \biggr)\beta _{1}^{\frac{p^{-}}{2}}, \quad \beta _{1}= \biggl(\frac{k}{B^{2}_{1}} \biggr)^{\frac{2}{p^{-}-2}}, $$
(9)
Define
where \((g\diamond u)(t)=\int _{0}^{t}g(t-\tau )\|u(x,\tau )- u(x,t)\| ^{2}_{L^{2}(\varOmega )}\,d\tau \).
$$ \begin{aligned}[b] E(t)={} &\frac{1}{2} \int _{\varOmega } \bigl\vert u_{t}(x,t) \bigr\vert ^{2}\,dx+\frac{1}{2} \biggl(1- \int _{0}^{t}g(\tau )\,d\tau \biggr) \int _{\varOmega } \bigl\vert \nabla u(x,t) \bigr\vert ^{2}\,dx \\ & {} +\frac{1}{2} \int _{0}^{t}g(t-\tau ) \bigl\Vert \nabla u(x, \tau )-\nabla u(x,t) \bigr\Vert ^{2}_{L^{2}(\varOmega )}\,d\tau - \int _{\varOmega }\frac{1}{p(x,t)} \vert u \vert ^{p(x,t)}(x,t)\,dx, \end{aligned} $$
(10)
Our main result is as follows.
Theorem 1.2
Assume that
\((H_{1})\)–\((H_{3})\)of Theorem 1.1hold, and that the following conditions are satisfied:
Then the solution of Problem (1) blows up in finite time
\(T^{*}\)satisfying the following estimate:
where the coefficients
\(C_{4}\), \(F(0)\), \(C_{5}\)are defined in (31) and (34), respectively, the exponents
\(q=p^{+}-1>1\), \(0<\lambda \leqslant \frac{p^{-}-m^{+}}{(m^{+}-1)p^{-}}\), \(J(0)=\int _{\varOmega }|u _{0}|^{p^{+}}\,dx\)and the coefficients
\(C_{6}\), \(C_{7}\)are defined in (42).
$$\begin{aligned} &(H_{4})\quad E(0)+\frac{ \vert \varOmega \vert }{p^{-}}< E_{1},\qquad \Vert \nabla u_{0} \Vert _{2} ^{2}> \frac{k}{B_{1}^{2}}\beta _{1}, \\ &(H_{5})\quad m^{+} < p^{-}, \\ &(H_{6})\quad \textit{there exists a sufficiently small }0< \varepsilon _{0}< 1\textit{ such that }1- \varepsilon _{0}\leqslant k< 1. \end{aligned}$$
$$\begin{aligned} \int ^{\infty }_{J(0)}\frac{1}{C_{6}y^{q}+y+C_{7}}\,dy\leqslant T^{*} \leqslant \frac{C_{5}(1-\lambda )}{F^{\frac{\lambda }{1-\lambda }}(0)C _{4}\lambda }, \end{aligned}$$
2 Proof of Theorem 1.2
In order to prove Theorem 1.2, we need the following lemmas.
Lemma 2.1
Proof
Following the lines of the proof of Lemma 2.1 in [18], we get \(E(t)\in C[0,T)\cap C^{1}(0,T)\) and
The above identity and conditions \(g(t)>0\), \(g'(t)<0\) show that
Next, we estimate the value of J.
In the second inequality of (13), we have used the following facts:
Inequality (11) follows from (12) and (13). □
$$\begin{aligned} E^{\prime }(t)={} &{-} \int _{\varOmega } \vert u_{t} \vert ^{m(x,t)}\,dx-\frac{g(t)}{2} \int _{\varOmega } \vert \nabla u \vert ^{2}\,dx- \int _{\varOmega }\frac{p_{t}}{p^{2}} \vert u \vert ^{p} \bigl(\ln \vert u \vert ^{p}-1 \bigr)\,dx \\ &{}+\frac{1}{2} \int _{0}^{t}g'(t-\tau ) \int _{\varOmega } \bigl\vert \nabla \bigl(u(\cdot,\tau )-u(\cdot,t)\bigr) \bigr\vert ^{2}\,dx\,d \tau . \end{aligned}$$
$$\begin{aligned} E'(t)+ \int _{\varOmega } \vert u_{t} \vert ^{m(x,t)}\leqslant - \int _{\varOmega }\frac{p _{t}}{p^{2}} \vert u \vert ^{p} \bigl(\ln \vert u \vert ^{p}-1 \bigr)\,dx:=J. \end{aligned}$$
(12)
$$ \begin{aligned}[b] J&\leqslant - \int _{\{ \vert u \vert ^{p}\leqslant e\}} \frac{ \vert u \vert ^{p(x,t)}}{p^{2}(x,t)} \bigl(\ln \vert u \vert ^{p(x,t)}-1 \bigr)p_{t}(x,t)\,dx \\ &\leqslant \int _{\{ \vert u \vert ^{p}\leqslant e\}} \frac{p_{t}(x,t)}{p^{2}(x,t)}\,dx\leqslant \int _{\varOmega }\frac{p_{t}(x,t)}{p^{2}(x,t)}\,dx. \end{aligned} $$
(13)
$$\begin{aligned} -\frac{1}{e}\leqslant s\ln s\leqslant 0, \quad 0\leqslant s\leqslant 1. \end{aligned}$$
Due to the lack of homogeneity and the existence of the gap between the norm and the modular, the control function constructed by Messaoudi in [18] fails in our problem, so we have to look for a new control function to establish the relations between the term \({\int _{\varOmega }\frac{u^{p(x,t)}(x,t)}{p(x,t)}\,dx}\) and the value of \({E_{1}}\), the following lemma helps us solve the problem.
Lemma 2.2
Assume that
\((H_{4})\)and
\(p^{-}>2\)hold, then there exists a positive constant
\(\beta _{2}>\beta _{1}\)such that
and
$$ k \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2}+(g \diamond \nabla u) (t)\geq \frac{k}{B_{1} ^{2}}\beta _{2}, \quad \forall t\geq 0 $$
(14)
$$ \int _{\varOmega }\frac{1}{p(x,t)}u^{p(x,t)}(x,t)\,dx \geq \frac{1}{p ^{-}}\max \bigl\{ \beta _{2}^{\frac{p^{+}}{2}}, \beta _{2}^{\frac{p^{-}}{2}} \bigr\} . $$
(15)
Proof
We borrow some ideas from [10]–[9]. First, we can derive from (7), (8), and (10) that
where \(\beta (t)= \frac{B^{2}_{1}}{k} [k\|\nabla u(t)\|_{2}^{2}+(g\diamond \nabla u)(t) ]\).
$$ \begin{aligned}[b] E(t) &\geq \frac{1}{2} \biggl[\biggl(1- \int _{0}^{t}g(s)\,ds\biggr) \bigl\Vert \nabla u(t) \bigr\Vert _{2} ^{2}+(g\diamond \nabla u) (t) \biggr]-\frac{1}{p^{-}} \int _{\varOmega }u^{p(x,t)}(\cdot,t) \,dx \\ &\geq \frac{1}{2} \bigl[k \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2}+(g\diamond \nabla u) (t) \bigr] - \frac{1}{p^{-}}\max \bigl\{ \bigl\Vert u(\cdot,t) \bigr\Vert _{p(\cdot),\varOmega }^{p^{+}}, \bigl\Vert u(\cdot,t) \bigr\Vert _{p(\cdot),\varOmega }^{p^{-}}\bigr\} \\ &\geq \frac{1}{2} \bigl[k \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2}+(g\diamond \nabla u) (t) \bigr] - \frac{1}{p^{-}}\max \bigl\{ B_{1}^{p^{+}} \Vert \nabla u \Vert _{2}^{p^{+}},B _{1}^{p^{-}} \Vert \nabla u \Vert _{2}^{p^{-}}\bigr\} \\ &=\frac{k}{2B_{1}^{2}}\beta -\frac{1}{p^{-}}\max \bigl\{ \beta ^{\frac{p^{+}}{2}},\beta ^{\frac{p^{-}}{2}}\bigr\} :=h\bigl(\beta (t)\bigr), \end{aligned} $$
(16)
Next, we analyze the properties of the function \(h(\beta )\). By calculating directly, we know that \(h(\beta )\) satisfies the following properties:
$$ \begin{aligned} &h(\beta )\in C[0,+\infty ); \\ &h'(\beta )= \textstyle\begin{cases} \frac{k}{2B_{1}^{2}}-\frac{p^{+}}{2p^{-}}\beta ^{\frac{p^{+}-2}{2}}< 0, &\beta >1; \\ \frac{k}{2B_{1}^{2}}-\frac{1}{2}\beta ^{\frac{p^{-}-2}{2}}, &0< \beta < 1; \end{cases}\displaystyle \\ &h'_{+}(1)=\frac{k}{2B_{1}^{2}}- \frac{p^{+}}{2p^{-}}< 0, \qquad h_{-}'(1)= \frac{k}{2B _{1}^{2}}-\frac{1}{2}< 0; \\ &h'(\beta _{1})=0, \quad 0< \beta _{1}= \biggl(\frac{k}{B_{1}^{2}} \biggr)^{\frac{2}{p^{-}-2}}< 1. \end{aligned} $$
(17)
Although the function \(h(\beta )\) is not differentiable at \(\beta =1\), a simple analysis shows that \(h(\beta )\) is increasing for \(0<\beta < \beta _{1}\), while \(h(\beta )\) is decreasing for \(\beta \geqslant \beta _{1}\), and \(\lim_{\beta \rightarrow \infty }h(\beta )=-\infty \). Due to \(E(0)+\frac{|\varOmega |}{p^{-}}< E_{1}\), there exists a positive constant \(\beta _{2}>\beta _{1}\) such that \(h(\beta _{2})=E(0)+\frac{| \varOmega |}{p^{-}}\). Let \(\beta _{0}=\|\nabla u_{0}\|_{2}^{2}\), then we have \(h(\frac{B^{2}_{1}\beta _{0}}{k})\leq E(0)+ \frac{|\varOmega |}{p^{-}}=h(\beta _{2})\). By (11) with \(\frac{B^{2}_{1}\beta _{0}}{k}>\beta _{1}\), we have \(\beta _{2}>\beta _{1}\).
To prove (14), we suppose that \(k\|\nabla u(t_{0})\|_{2} ^{2}+(g\diamond \nabla u)(t_{0})<\frac{k}{B_{1}^{2}}\beta _{2}\) for some \(t_{0}>0\). By the continuity of \(k\|\nabla u(t)\|_{2}^{2}+(g\diamond \nabla u)(t)\), we may choose \(t_{1}>0\) such that
And then, combining the monotonicity of \(h(\beta )\) with the above inequalities, we have
which contradicts (11).
$$ \frac{k}{B_{1}^{2}}\beta _{1}< k \bigl\Vert \nabla u(t_{1}) \bigr\Vert _{2}^{2}+(g\diamond \nabla u) (t_{1})< \frac{k}{B_{1}^{2}}\beta _{2}. $$
$$ E(0)+\frac{ \vert \varOmega \vert }{p^{-}}=h(\beta _{2})< h \biggl( \frac{B_{1}^{2}}{k} \bigl(k \bigl\Vert \nabla u(t_{1}) \bigr\Vert _{2}^{2}+(g\diamond \nabla u) (t_{1}) \bigr) \biggr)\leq E(t_{1}), $$
From (10) we can see that
which implies that
□
$$ \frac{1}{2} \bigl[k \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2}+(g\diamond \nabla u) (t) \bigr]\leq E(0)+ \frac{ \vert \varOmega \vert }{p^{-}}+ \int _{\varOmega }\frac{1}{p(x,t)}u ^{p(x,t)}(t)\,dx, $$
$$\begin{aligned} \int _{\varOmega }\frac{1}{p(x,t)}u^{p(x,t)}(x,t)\,dx &\geq \frac{1}{2} \bigl[k \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2}+(g\diamond \nabla u) (t) \bigr]-E(0)- \frac{ \vert \varOmega \vert }{p^{-}} \\ &\geq \frac{k}{2B_{1}^{2}}\beta _{2}-h(\beta _{2})= \frac{1}{p^{-}} \max \bigl\{ \beta _{2}^{\frac{p^{+}}{2}},\beta _{2}^{\frac{p^{-}}{2}}\bigr\} . \end{aligned}$$
Let \(H(t)=E_{1}-E(t)\), \(t\geq 0 \), we have the following.
Lemma 2.3
For all
\(t>0\),
$$ 0< H(0)-\frac{ \vert \varOmega \vert }{p^{-}}< H(t)\leq \int _{\varOmega } \frac{u^{p(x,t)}(x,t)}{p(x,t)}\,dx. $$
(18)
Proof
By Lemma 2.1, we have \(H^{\prime }(t)=-E'(t)\). Inequalities (12) and (13) show that
The left inequality in (18) follows from (19) and \(E(0)+\frac{|\varOmega |}{p^{-}}< E_{1}\).
$$\begin{aligned} H'(t)\geqslant - \int _{\varOmega }\frac{p_{t}(x,t)}{p^{2}(x,t)}\,dx. \end{aligned}$$
(19)
On the other hand, (10) and (15) yield
From (11) and (13), it is easy to verify that
This completes the proof of Lemma 2.3. □
$$\begin{aligned} H(t)={}&E_{1}-\frac{1}{2} \int _{\varOmega } \bigl\vert u_{t}(x,t) \bigr\vert ^{2}\,dx-\frac{1}{2} \biggl(1- \int _{0}^{t}g(\tau )\,d\tau \biggr) \int _{\varOmega } \bigl\vert \nabla u(x,t) \bigr\vert ^{2}\,dx \\ &{}-\frac{1}{2} \int _{0}^{t}g(t-\tau ) \bigl\Vert \nabla u(x, \tau )-\nabla u(x,t) \bigr\Vert ^{2}_{L^{2}(\varOmega )}\,d\tau + \int _{\varOmega }\frac{1}{p(x,t)} \vert u \vert ^{p(x,t)}(x,t)\,dx. \end{aligned}$$
$$\begin{aligned} &E_{1}-\frac{1}{2} \int _{\varOmega } \bigl\vert u_{t}(x,t) \bigr\vert ^{2}\,dx-\frac{1}{2} \biggl(1- \int _{0}^{t}g(\tau )\,d\tau \biggr) \int _{\varOmega } \bigl\vert \nabla u(x,t) \bigr\vert ^{2}\,dx \\ &\qquad {}-\frac{1}{2} \int _{0}^{t}g(t-\tau ) \bigl\Vert \nabla u(x, \tau )-\nabla u(x,t) \bigr\Vert ^{2}_{L^{2}(\varOmega )}\,d\tau \\ &\quad \leq E_{1}-\frac{\alpha _{2}}{2}\leq E_{1}- \frac{\alpha _{1}}{2}\leq 0, \quad t>0. \end{aligned}$$
Proof of Theorem 1.2
This proof will be divided into two steps.
Step 1. The idea of this proof mainly comes from [18]. Choose \(0<\lambda \leqslant \min \{\frac{p^{-}-m}{(m-1)p^{-}}, \frac{p ^{-}-2}{2p^{-}}\}< \frac{1}{2}\) and define \(F(t)=H^{1-\lambda }(t)+\varepsilon \int _{ \varOmega }u_{t}(x,t)u(x,t)\,dx\). Then
According to the first identity in Problem (1) and Cauchy inequality with δ, we obtain the following inequalities:
where the coefficient δ will be determined later.
$$ \begin{aligned}[b] F'(t) ={}&(1-\lambda )H^{-\lambda }(t)H'(t)+\varepsilon \int _{\varOmega }u _{tt}(x,t)u(x,t)\,dx+\varepsilon \int _{\varOmega }u^{2}_{t}(x,t)\,dx \\ ={}&(1-\lambda )H^{-\lambda }(t) \biggl[ \int _{\varOmega } \vert u_{t} \vert ^{m(x,t)}\,dx+ \frac{g(t)}{2} \bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2}-\frac{1}{2}\bigl(g' \diamond \nabla u\bigr) (t) \biggr] \\ & {} + \varepsilon \int _{\varOmega }u_{tt}u\,dx+\varepsilon \int _{\varOmega }u^{2} _{t}\,dx:=J_{1}+J_{2}+J_{3}. \end{aligned} $$
(20)
$$\begin{aligned} J_{2} =&\varepsilon \int _{\varOmega }u_{tt}u\,dx=\varepsilon \int _{\varOmega }u \biggl(\Delta u- \int _{0}^{t}g(t-\tau )\Delta u(\tau )\,d\tau - \vert u_{t} \vert ^{m(x,t)-2}u _{t}+ \vert u \vert ^{p(x,t)-2}u \biggr)\,dx \\ =&\varepsilon \biggl[-\biggl(1- \int _{0}^{t}g(s)\,ds\biggr) \int _{\varOmega } \vert \nabla u \vert ^{2}\,dx+ \int _{0}^{t}g(t-\tau ) \int _{\varOmega }\bigl(\nabla u(\tau )-\nabla u(t)\bigr) \nabla u(t)\,dx\,d\tau \\ & {} - \int _{\varOmega } \vert u_{t} \vert ^{m(x,t)-2}u_{t}u\,dx+ \int _{\varOmega } \vert u \vert ^{p(x,t)}\,dx \biggr] \\ \geqslant& -\varepsilon \biggl[1- \int _{0}^{t}g(s)\,ds+\frac{1}{4\delta } \int _{0}^{t}g(s)\,ds \biggr] \int _{\varOmega } \vert \nabla u \vert ^{2}\,dx- \varepsilon \int _{\varOmega } \vert u_{t} \vert ^{m(x,t)-2}u_{t}u\,dx \\ & {} +\varepsilon \int _{\varOmega } \vert u \vert ^{p(x,t)}\,dx -\varepsilon \delta \int _{0}^{t}g(t-\tau ) \int _{\varOmega } \bigl\vert \nabla \bigl(u(\tau )-\nabla u(t) \bigr) \bigr\vert ^{2}\,dx\,d \tau , \end{aligned}$$
(21)
Moreover, using the condition \(g(t)>0\), \(g'(t)<0\), we have
$$ \begin{aligned} J_{1}\geqslant (1-\lambda )H^{-\lambda }(t) \int _{\varOmega } \vert u_{t} \vert ^{m(x,t)}\,dx. \end{aligned} $$
(22)
Applying Young’s inequality with \(\eta >1\) and the conditions \(m^{+}< p^{-}\), \(0<\lambda \leqslant \frac{p^{-}-m^{+}}{(m^{+}-1)p^{-}}\) and Lemma 2.3, we have
where η will be determined later and the constants are defined as follows:
$$ \begin{aligned}[b] & \biggl\vert \int _{\varOmega }|u_{t}| ^{m(x,t)-1}|u | \,dx \biggr\vert \\ &\quad \leqslant \eta ^{\frac{m^{-}}{m^{-}-1}}H^{-\lambda }(t) \int _{\varOmega } \vert u_{t} \vert ^{m(x,t)}\,dx+\frac{1}{ \eta ^{m^{-}}}C^{{\lambda (m^{-}-m^{+})}}_{1} H^{\lambda (m^{+}-1)}(t) \int _{\varOmega } \vert u \vert ^{m(x,t)}\,dx \\ &\quad \leqslant \eta ^{\frac{m^{-}}{m^{-}-1}}H^{-\lambda }(t) \int _{\varOmega } \vert u _{t} \vert ^{m(x,t)}\,dx+\frac{C_{2}}{\eta ^{m^{-}}}H^{\lambda (m^{+}-1)}(t) \max \bigl\{ \Vert u \Vert ^{m^{+}}_{p(\cdot)}, \Vert u \Vert ^{m^{-}}_{p(\cdot)} \bigr\} , \end{aligned} $$
(23)
$$\begin{aligned} C_{1}:=\min \biggl\{ \biggl(H(0)-\frac{ \vert \varOmega \vert }{p^{-}}\biggr),1 \biggr\} , \qquad C_{2}=\bigl(1+ \vert \varOmega \vert \bigr)^{m^{+}}C^{{\lambda (m^{-}-m^{+})}}_{1}. \end{aligned}$$
According to (18) and (7), we have
$$ \begin{aligned} \Vert u \Vert ^{m^{+}}_{p(\cdot)} &\leqslant \max \biggl\{ \biggl( \int _{\varOmega } \vert u \vert ^{p(x,t)}\,dx \biggr)^{\frac{m ^{+}}{p^{+}}},\biggl( \int _{\varOmega } \vert u \vert ^{p(x,t)}\,dx \biggr)^{\frac{m^{+}}{p^{-}}} \biggr\} \\ &\leqslant \max \bigl\{ 1,H^{\frac{m^{+}}{p^{+}}-\frac{m^{+}}{p^{-}}}(t) \bigr\} \biggl( \int _{\varOmega } \vert u \vert ^{p(x,t)}\,dx \biggr)^{\frac{m^{+}}{p^{-}}}; \\ \Vert u \Vert ^{m^{-}}_{p(\cdot)} &\leqslant \max \biggl\{ \biggl( \int _{\varOmega } \vert u \vert ^{p(x,t)}\,dx \biggr)^{\frac{m ^{-}}{p^{+}}},\biggl( \int _{\varOmega } \vert u \vert ^{p(x,t)}\,dx \biggr)^{\frac{m^{-}}{p^{-}}} \biggr\} \\ &\leqslant \max \bigl\{ H^{\frac{m^{-}-m^{+}}{p^{-}}}(t),H^{\frac{m ^{-}}{p^{+}}-\frac{m^{+}}{p^{-}}}(t) \bigr\} \biggl( \int _{\varOmega } \vert u \vert ^{p(x,t)}\,dx \biggr)^{\frac{m ^{+}}{p^{-}}}. \end{aligned} $$
(24)
From (18) and (24), it follows easily
where \(C_{3}=2C^{\frac{m^{-}}{p^{+}}-\frac{m^{+}}{p^{-}}}_{1}\).
$$\begin{aligned} \max \bigl\{ \Vert u \Vert ^{m^{+}}_{p(\cdot)}, \Vert u \Vert ^{m^{-}}_{p(\cdot)} \bigr\} \leqslant C_{3} \biggl( \int _{\varOmega } \vert u \vert ^{p(x,t)}\,dx \biggr)^{\frac{m^{+}}{p^{-}}}, \end{aligned}$$
(25)
Furthermore, by (20)–(22) and (25), we have
$$ \begin{aligned}[b] F'(t)\geqslant{}& \bigl(1-\lambda -\varepsilon \eta ^{\frac{m^{-}}{m^{-}-1}}\bigr)H ^{-\lambda }(t) \int _{\varOmega } \vert u_{t} \vert ^{m(x,t)}\,dx+\varepsilon \int _{ \varOmega }u^{2}_{t}\,dx\\ &{}+\varepsilon \int _{\varOmega } \vert u \vert ^{p(x,t)}\,dx -\varepsilon \delta (g\diamond \nabla u) (t)) \\ & {} -\varepsilon \biggl[1- \int _{0}^{t}g(s)\,ds+\frac{1}{4\delta } \int _{0}^{t}g(s)\,ds \biggr] \int _{\varOmega } \vert \nabla u \vert ^{2}\,dx\\ &{}- \frac{\varepsilon C_{2}C_{3}C^{ \lambda (m^{+}-1)+\frac{m^{+}}{p^{-}}-1}_{1}}{\eta ^{m^{-}}} \int _{ \varOmega } \vert u \vert ^{p(x,t)}\,dx. \end{aligned} $$
(26)
Utilizing the definition of \(H(t)\) and \(E(t)\), we get
Inequality (15) and the definition of \(E_{1}\) show that the following inequality holds:
Again choosing \(2<\theta <\frac{2p^{-}\max \{\beta _{2}^{ \frac{p^{+}}{2}},\beta _{2}^{\frac{p^{-}}{2}}\}}{2\max \{\beta _{2} ^{\frac{p^{+}}{2}}, \beta _{2}^{\frac{p^{-}}{2}}\}+(p^{-}-2)\beta _{1} ^{\frac{p^{-}}{2}}}<p^{-}\) and then applying (26)–(28), we have
$$\begin{aligned} \begin{aligned}[b] &\int _{\varOmega }\frac{1}{p(x,t)} \vert u \vert ^{p(x,t)}\,dx\\ &\quad =H(t)-E_{1}+\frac{1}{2} \biggl[ \Vert u_{t} \Vert _{2}^{2} +\biggl(1- \int _{0}^{t}g(s)\,ds\biggr) \int _{\varOmega } \bigl\vert \nabla u(t) \bigr\vert ^{2}\,dx+(g \diamond \nabla u) (t) \biggr]. \end{aligned} \end{aligned}$$
(27)
$$\begin{aligned} E_{1}\leqslant \frac{(p^{-}-2)\beta _{1}^{\frac{p^{-}}{2}}}{2\max \{ \beta _{2}^{\frac{p^{+}}{2}}, \beta _{2}^{\frac{p^{-}}{2}}\}} \int _{ \varOmega }\frac{1}{p(x,t)} \vert u \vert ^{p(x,t)}\,dx. \end{aligned}$$
(28)
$$\begin{aligned} F'(t) \geqslant& \bigl[1-\lambda -\varepsilon \eta ^{ \frac{m^{-}}{m^{-}-1}} \bigr]H^{-\lambda }(t) \int _{\varOmega } \vert u_{t} \vert ^{m(x,t)}\,dx+ \varepsilon \biggl(1+\frac{\theta }{2}\biggr) \Vert u_{t} \Vert ^{2}_{2}+\varepsilon \theta H(t) \\ &{} +\varepsilon \biggl(\frac{\theta }{2}-\delta \biggr) (g\diamond \nabla u) (t)+ \varepsilon \biggl(\frac{\theta -2}{2}-\biggl(\frac{\theta -2}{2}+ \frac{1}{4 \delta }\biggr) \int _{0}^{\infty }g(s)\,ds \biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2} \\ & {} +\varepsilon \biggl(p^{-}-\theta -\frac{p^{+} C_{2}C_{3}C^{\lambda (m ^{+}-1)+\frac{m^{+}}{p^{-}}-1}_{1}}{\eta ^{m^{-}}} \\ &{}- \frac{(p^{-}-2) \beta _{1}^{\frac{p^{-}}{2}}\theta }{2\max \{\beta _{2}^{ \frac{p^{+}}{2}}, \beta _{2}^{\frac{p^{-}}{2}}\}} \biggr) \int _{\varOmega } \frac{1}{p(x,t)} \vert u \vert ^{p(x,t)}. \end{aligned}$$
(29)
Choosing δ, η, ε such that
and dropping nonnegative terms, in which we use condition \((H_{6})\), we have
where
$$\begin{aligned} &0< \delta < \frac{\theta }{2},\qquad \frac{p^{+} C_{2}C_{3}C^{\lambda (m^{+}-1)+\frac{m ^{+}}{p^{-}}-1}_{1}}{\eta ^{m^{-}}}< p^{-}-\theta -\frac{(p^{-}-2)\beta _{1}^{\frac{p^{-}}{2}}\theta }{2\max \{\beta _{2}^{\frac{p^{+}}{2}}, \beta _{2}^{\frac{p^{-}}{2}}\}},\\ & 0< \varepsilon < (1-\lambda ) \eta ^{\frac{m^{-}}{1-m^{-}}} \end{aligned}$$
$$\begin{aligned} F'(t) &\geqslant C_{4} \biggl[ \Vert u_{t} \Vert ^{2}_{2}+H(t)+(g \diamond \nabla u) (t)+ \int _{\varOmega } \vert u \vert ^{p(x,t)}\,dx \biggr], \end{aligned}$$
(30)
$$ \begin{aligned} &C_{4}=\min \biggl\{ \varepsilon \biggl(1+\frac{\theta }{2}\biggr),\varepsilon \theta , \varepsilon \biggl(\frac{\theta }{2}-\delta \biggr), \varepsilon \biggl(p ^{-}-\theta -\frac{C_{2}p^{+}}{\eta ^{m}}-\frac{(p^{-}-2)\beta _{1}^{\frac{p ^{-}}{2}}\theta }{2\max \{\beta _{2}^{\frac{p^{+}}{2}}, \beta _{2}^{\frac{p ^{-}}{2}}\}} \biggr) \frac{1}{p^{+}} \biggr\} ; \\ &F(0)=H^{1-\lambda }(0)+\varepsilon \int _{\varOmega }u_{0}u_{1}\,dx>0. \end{aligned} $$
(31)
Applying Hölder’s inequality, the embedding \(L^{p(\cdot)}( \varOmega )\hookrightarrow L^{p^{-}}(\varOmega )\hookrightarrow L^{2}( \varOmega )\) (\(p^{-}>2\)), Young’s inequality, and inequality (7), we have that
$$ \begin{aligned}[b] &\biggl( \biggl\vert \int _{\varOmega }u_{t}u\,dx \biggr\vert \biggr)^{\frac{1}{1-\lambda }} \\ &\quad \leqslant \bigl( \Vert u \Vert _{2} \Vert u_{t} \Vert _{2}\bigr)^{\frac{1}{1-\lambda }} \leqslant \bigl(1+ \vert \varOmega \vert \bigr)^{\frac{1}{(1- \lambda )}} \Vert u \Vert ^{\frac{1}{1-\lambda }}_{p(\cdot)} \Vert u_{t} \Vert ^{\frac{1}{1- \lambda }}_{2} \\ &\quad \leqslant \bigl(1+ \vert \varOmega \vert \bigr)^{\frac{1}{(1-\lambda )}} \bigl[ \Vert u \Vert ^{\frac{2}{1-2 \lambda }}_{p(x,t)}+ \Vert u_{t} \Vert ^{2}_{2} \bigr] \\ &\quad \leqslant \bigl(1+ \vert \varOmega \vert \bigr)^{\frac{1}{(1-\lambda )}} \max \biggl\{ \biggl( \int _{\varOmega } \vert u \vert ^{p(x,t)}\,dx \biggr)^{\frac{2}{(1-2\lambda )p^{-}}},\biggl( \int _{\varOmega } \vert u \vert ^{p(x,t)}\,dx \biggr)^{\frac{2}{(1-2\lambda )p^{+}}} \biggr\} \\ &\qquad {}+\bigl(1+ \vert \varOmega \vert \bigr)^{\frac{1}{(1-\lambda )}} \Vert u_{t} \Vert ^{2}_{2} \\ &\quad \leqslant \bigl(1+ \vert \varOmega \vert \bigr)^{\frac{1}{(1-\lambda )}}C^{\frac{2-(1-2 \lambda )p^{+}}{(1-2\lambda )p^{+}}}_{1} \int _{\varOmega } \vert u \vert ^{p(x,t)}\,dx +\bigl(1+ \vert \varOmega \vert \bigr)^{\frac{1}{(1-\lambda )}} \Vert u_{t} \Vert ^{2}_{2}. \end{aligned} $$
(32)
By (32) and the definition of \(F(t)\), we get
with
$$ \begin{aligned}[b] F^{\frac{1}{1-\lambda }}(t)\leqslant{}& 2^{\frac{1}{1-\lambda }} \biggl[H(t)+ \varepsilon ^{\frac{1}{1-\lambda }} \biggl\vert \int _{\varOmega }u_{t}u\,dx \biggr\vert ^{\frac{1}{1- \lambda }} \biggr] \\ \leqslant{}& 2^{\frac{1}{1-\lambda }} \biggl[H(t)+\bigl(1+ \vert \varOmega \vert \bigr)^{\frac{1}{(1- \lambda )}}C^{\frac{2-(1-2\lambda )p^{+}}{(1-2\lambda )p^{+}}}_{1} \int _{\varOmega } \vert u \vert ^{p(x,t)}\,dx \\ &{}+\bigl(1+ \vert \varOmega \vert \bigr)^{\frac{1}{(1-\lambda )}} \Vert u_{t} \Vert ^{2}_{2} \biggr] \\ \leqslant{}& C_{5} \biggl[H(t)+ \int _{\varOmega } \vert u \vert ^{p(x,t)}\,dx + \Vert u_{t} \Vert ^{2}_{2} \biggr] \end{aligned} $$
(33)
$$\begin{aligned} C_{5}=2^{\frac{1}{1-\lambda }} \bigl(1+\bigl(1+ \vert \varOmega \vert \bigr)^{\frac{1}{(1- \lambda )}}C^{\frac{2-(1-2\lambda )p^{+}}{(1-2\lambda )p^{+}}}_{1} + \bigl(1+ \vert \varOmega \vert \bigr)^{\frac{1}{(1-\lambda )}} \bigr). \end{aligned}$$
(34)
By (30) and (33), we obtain the following inequality:
$$\begin{aligned} F'(t)\geqslant \frac{C_{4}}{C_{5}}F^{\frac{1}{1-\lambda }}(t). \end{aligned}$$
(35)
By Gronwall’s inequality, we have
Therefore, inequality (36) implies that \(F(t)\) blows up in finite time
$$\begin{aligned} F^{\frac{\lambda }{1-\lambda }}(t)\geqslant \frac{1}{F^{\frac{-\lambda }{1-\lambda }}(0)-\frac{C_{4}\lambda }{C_{5}(1-\lambda )}t}. \end{aligned}$$
(36)
$$\begin{aligned} T^{*}\leqslant \frac{C_{5}(1-\lambda )}{F^{\frac{\lambda }{1-\lambda }}(0)C_{4}\lambda }. \end{aligned}$$
(37)
Step 2. We give a lower bound for blow-up time \(T^{*}\).
Define \(J(t)=\int _{\varOmega }|u|^{p^{+}}\,dx\), then
By \(2< p^{+}\leqslant \frac{2N-2}{N-2}\) and (8), one has
$$\begin{aligned} J'(t)=p^{+} \int _{\varOmega } \vert u \vert ^{p^{+}-2}uu_{t}\,dx \leqslant p^{+} \biggl[ \int _{\varOmega } \vert u \vert ^{2p^{+}-2}\,dx+ \int _{\varOmega } \vert u_{t} \vert ^{2}\,dx \biggr]. \end{aligned}$$
(38)
$$\begin{aligned} \int _{\varOmega } \vert u \vert ^{2p^{+}-2}\,dx\leqslant B^{2p^{+}-2} \Vert \nabla u \Vert ^{2p ^{+}-2}_{2}. \end{aligned}$$
(39)
Recalling the definition of \(E(t)\) and \(E(t)\leqslant E_{1}\), we get
$$ \begin{aligned}[b] k \Vert \nabla u \Vert ^{2}_{2}+ \Vert u_{t} \Vert ^{2}_{2} &\leqslant E_{1}+ \int _{\varOmega } \vert u \vert ^{p(x,t)} \,dx\\ &\leqslant E_{1}+ \int _{\{ \vert u \vert \geqslant 1\}} \vert u \vert ^{p(x,t)}\,dx+ \int _{\{ \vert u \vert \leqslant 1\}} \vert u \vert ^{p(x,t)}\,dx \\ &\leqslant E_{1}+ \int _{\varOmega } \vert u \vert ^{p^{+}}\,dx+ \vert \varOmega \vert . \end{aligned} $$
(40)
Combining (38)–(40) with the inequality \((|a|+|b|)^{q}\leqslant 2^{q-1}(|a|^{q}+|b|^{q})\) (\(q>1\)), we have
with
$$\begin{aligned} J'(t)&\leqslant C_{6}k^{1-p^{+}}J^{q}(t)+C_{6} \biggl(\frac{ E_{1}+ \vert \varOmega \vert }{k}\biggr)^{p ^{+}-1}+ E_{1}+ \vert \varOmega \vert +J(t) \\ &\leqslant C_{6}k^{1-p^{+}}J^{q}(t)+J(t)+C _{7}, \end{aligned}$$
(41)
$$\begin{aligned} C_{6}=2^{p^{+}-2}p^{+}B^{2p^{+}-2}, \qquad C_{7}=C_{6}\biggl(\frac{E_{1}+ \vert \varOmega \vert }{k} \biggr)^{p^{+}-1}+ E_{1}+ \vert \varOmega \vert . \end{aligned}$$
(42)
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