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Blow-up and global existence for solution of quasilinear viscoelastic wave equation with strong damping and source term
- Jianghao Hao1Email authorView ORCID ID profile and
- Haiying Wei1
- Received: 10 February 2017
- Accepted: 21 April 2017
- Published: 5 May 2017
Abstract
In this paper we consider a quasilinear viscoelastic wave equation with initial-boundary conditions, strong damping and source term. Under suitable assumptions on the initial data and the relaxation function, we establish a blow-up result of a solution for negative initial energy and some positive initial energy if the influence of the source term is greater than the dissipation. We show that the solution exists globally for any initial data if the influence of dissipation is greater than the source term.
Keywords
- viscoelasticity wave equation
- strong damping
- blow-up
- global existence
1 Introduction
In this work, we study the following quasilinear viscoelastic wave equation with initial-boundary value conditions, strong damping and source term:
where Ω is a bounded domain of \(R^{n}\) (\(n\geq1\)) with smooth boundary ∂Ω, \(\rho>0\) and \(p>2\) are constants. The relaxation function g is a given function to be specified later.
$$ \textstyle\begin{cases} \vert u_{t}\vert ^{\rho}u_{tt}-\triangle u+ {\int_{0}^{t} g(t-\tau) \triangle u(\tau)\,d\tau}-\triangle u_{t}=\vert u\vert ^{p-2}u, & \mbox{in } \Omega\times(0, \infty),\\ u(x,t)=0,& \mbox{on } \partial\Omega\times(0, \infty),\\ u(x,0)=u_{0}(x), \qquad u_{t}(x,0)=u_{1}(x), & \mbox{in } \bar{\Omega}, \end{cases} $$
(1.1)
As is well known, the wave equation with memory has been extensively studied. Berrimi and Messaoudi [1] considered the following initial-boundary value problem:
where Ω is a bounded domain of \(R^{n}\) (\(n\geq1\)) with smooth boundary ∂Ω, γ is a positive constant, and g is a nonnegative and decreasing function. They obtained a local existence result and proved, for certain initial data and suitable conditions on g and γ (under weaker conditions than those in [2, 3]), that the solution is global and decays uniformly (exponentially or polynomially depending on the decay rate of the relaxation function g) if the initial data is small enough. For further work on the existence and the decay of solutions, we refer the reader to [4–8]. Messaoudi [9] discussed the following initial-boundary value problem:
where Ω is a bounded domain of \(R^{n}\) (\(n\geq1\)) with smooth boundary ∂Ω, \(m\geq1\), \(p>2\), \(a,b>0\) are constants and \(g:R^{+}\rightarrow R^{+}\) is a positive nonincreasing function. Under suitable conditions on g, he proved that solutions with negative initial energy blow up in finite time if \(p>m\), and continue to exist if \(p\leq m\). For the same problem, Messaoudi [10] extended this result to certain solutions with initial positive energy. A similar result was also obtained by Lu and Li [11], Guo and Lin [12].
$$ \textstyle\begin{cases} u_{tt}-\triangle u+ \int_{0}^{t} g(t-\tau)\triangle u(\tau)\,d\tau=\vert u\vert ^{\gamma}u,& \mbox{in } \Omega\times(0, \infty),\\ u(x,t)=0,& \mbox{on } \partial\Omega\times(0, \infty),\\ u(x,0)=u_{0}(x), \qquad u_{t}(x,0)=u_{1}(x),& \mbox{in } \bar{\Omega}, \end{cases} $$
(1.2)
$$ \textstyle\begin{cases} u_{tt}-\triangle u+ {\int_{0}^{t} g(t-\tau)\triangle u(\tau)\,d\tau }+a\vert u_{t}\vert ^{m-2}u_{t}=b\vert u\vert ^{p-2}u,& \mbox{in } \Omega \times(0, \infty),\\ u(x,t)=0,& \mbox{on } \partial\Omega\times(0, \infty),\\ u(x,0)=u_{0}(x), \qquad u_{t}(x,0)=u_{1}(x),& \mbox{in } \bar{\Omega}, \end{cases} $$
(1.3)
Recently, Song and Zhong [13] studied a nonlinear viscoelastic problem with strong damping:
where Ω is a bounded domain of \(R^{n}\) (\(n\geq1\)) with smooth boundary ∂Ω, \(2< p<\frac{2n-2}{n-2}\). They proved that solutions with positive initial energy blow up in finite time using the potential well method introduced by Payne and Sattinger [14]. Furthermore, Song and Xue [15] extended this result to arbitrarily high initial energy.
$$ \textstyle\begin{cases} u_{tt}-\triangle u+ {\int_{0}^{t} g(t-\tau)\triangle u(\tau)\,d\tau }-\triangle u_{t}= \vert u\vert ^{p-2}u,& \mbox{in } \Omega\times(0, \infty),\\ u(x,t)=0,& \mbox{on } \partial\Omega\times(0, \infty),\\ u(x,0)=u_{0}(x), \qquad u_{t}(x,0)=u_{1}(x),& \mbox{in } \bar{\Omega}, \end{cases} $$
(1.4)
In the same direction Cavalcanti et al. [16] considered the following initial-boundary value problem:
where Ω is a bounded domain of \(R^{n}\) (\(n\geq1\)) with smooth boundary ∂Ω and \(\rho>0\). They proved a global existence result for \(\gamma\geq0\) and an exponential decay result for \(\gamma >0\). Cavalcanti et al. [17] studied problem (1.5) with \(\rho\geq0\) and \(\gamma\geq0\). The authors showed that the energy decays to zero uniformly with the rate that is determined from the solutions of the ODE quantifying the behavior of \(g(t)\), and they improved many previous results. In the case of \(\gamma=0\), Liu [18] discussed the following problem:
where Ω is a bounded domain of \(R^{n}\) (\(n\geq1\)) with smooth boundary ∂Ω, and \(\rho, b>0\), \(p>2\) are constants. He obtained a general decay of the solution for certain class of relaxation functions and initial data in the stable set, and showed that the solution blows up in a larger class of initial positive energy. Furthermore, Song [19] studied the following problem:
where Ω is a bounded domain of \(R^{n}\) (\(n\geq1\)) with smooth boundary ∂Ω, \(m>2\), \(g:R^{+}\rightarrow R^{+}\) is a positive nonincreasing function. He proved the nonexistence of global solution of (1.7) with initial positive energy.
$$ \textstyle\begin{cases} \vert u_{t}\vert ^{\rho}u_{tt}-\triangle u-\triangle u_{tt}+ {\int _{0}^{t} g (t-\tau)\triangle u(\tau)\,d\tau}-\gamma\triangle u_{t}=0,& \mbox{in } \Omega\times(0, \infty),\\ u(x,t)=0,& \mbox{on } \partial\Omega\times(0, \infty),\\ u(x,0)=u_{0}(x), \qquad u_{t}(x,0)=u_{1}(x),& \mbox{in } \bar{\Omega}, \end{cases} $$
(1.5)
$$ \textstyle\begin{cases} \vert u_{t}\vert ^{\rho}u_{tt}-\triangle u-\triangle u_{tt}+ {\int_{0}^{t} g(t-\tau)\triangle u(\tau)\,d\tau}=b\vert u\vert ^{p-2}u,& \mbox{in } \Omega\times(0, \infty),\\ u(x,t)=0,& \mbox{on } \partial\Omega\times(0, \infty),\\ u(x,0)=u_{0}(x), \qquad u_{t}(x,0)=u_{1}(x),& \mbox{in } \bar{\Omega}, \end{cases} $$
(1.6)
$$ \textstyle\begin{cases} \vert u_{t}\vert ^{\rho}u_{tt}-\triangle u+ {\int_{0}^{t} g(t-\tau )\triangle u(\tau) \,d\tau}+\vert u_{t}\vert ^{m-2}u_{t}=b\vert u\vert ^{p-2}u,& \mbox{in } \Omega\times[0, T],\\ u(x,t)=0,& \mbox{on } \partial\Omega\times[0, T],\\ u(x,0)=u_{0}(x), \qquad u_{t}(x,0)=u_{1}(x),& \mbox{in } \bar{\Omega}, \end{cases} $$
(1.7)
Motivated by the above pioneering work, we consider the problem (1.1). Under suitable assumptions on the initial data and the relaxation function g, we obtain a blow-up result for the solution with negative initial energy and some positive initial energy if \(p>\rho+2\), and get a global existence result for any initial data if \(p\leq\rho+2\) using the perturbed energy functional technique. This paper is organized as follows. In Section 2, we present some assumptions and preliminaries. Section 3 is devoted to the blow-up result. In Section 4, we obtain the global existence result.
2 Preliminaries
In this section, we shall give some notations and preliminaries used throughout this paper. Denote by \(\Vert \cdot \Vert _{p}\) the usual norm in \(L^{p}(\Omega)\) (\(p\geq2\)). Let B be the best embedding constant such that \(\Vert \phi \Vert _{p}\leq B \Vert \nabla\phi \Vert _{2}\), \(\phi\in H^{1}_{0}(\Omega)\). Besides, C and \(C_{i}\) (\(i\in N^{+}\)) denote general positive constants, which may be different in different estimates.
Now, we make the following assumptions.
- (G1)\(g(t)\): \(R^{+}\rightarrow R^{+}\) is a \(C^{1}\) function satisfies$$\begin{aligned}& g'(s)\leq0, \end{aligned}$$(2.1)$$\begin{aligned}& 1- \int_{0}^{\infty}g(s)\,ds=l>0. \end{aligned}$$(2.2)
- (G2)For the nonlinear term, we assume$$ \textstyle\begin{cases} 2< p< \infty, \quad \mbox{if } n=1, 2\quad \mbox{and}\quad 2< p\leq \frac{2(n-1)}{n-2}, \quad\mbox{if } n\geq3, \\ 0< \rho< \infty, \quad \mbox{if } n=1, 2 \quad \mbox{and} \quad 2< \rho \leq\frac{2}{n-2}, \quad \mbox{if } n\geq3. \end{cases} $$(2.3)
We first state, without a proof, a local existence theorem which can be established by the Faedo-Galerkin method. The interested reader can refer to Cavalcanti et al. [5] for details.
Theorem 2.1
Assume (G1) and (G2) hold and
\((u_{0}, u_{1})\in H^{1}_{0}(\Omega )\times L^{2}(\Omega)\)
is given. Then problem (1.1) has a unique local solution
$$\begin{aligned} u\in C\bigl([0,T];H^{1}_{0}(\Omega)\bigr),\qquad u_{t}\in C\bigl([0,T];H^{1}_{0}(\Omega)\bigr). \end{aligned}$$
(2.4)
Lemma 2.2
Assume (G1) and (G2) hold. Let
\(u(t)\)
be a solution of (1.1). Then
\(E(t)\)
is nonincreasing. Moreover, for
\(t>0\), the following inequality holds:
where
and
$$\begin{aligned} E'(t)=-\bigl\Vert \nabla u_{t}(t)\bigr\Vert ^{2}_{2}+\frac{1}{2}\bigl(g'\circ \nabla u\bigr) (t)-\frac{1}{2}g(t)\bigl\Vert \nabla u(t)\bigr\Vert ^{2}_{2}\leq0, \end{aligned}$$
(2.5)
$$E(t)=\frac{1}{\rho+2}\Vert u_{t}\Vert ^{\rho+2}_{\rho+2}+ \frac{1}{2}\biggl(1- \int_{0}^{t} g(s)\,ds\biggr)\bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2}+ \frac{1}{2}(g\circ\nabla u) (t)-\frac{1}{p}\bigl\Vert u(t)\bigr\Vert ^{p}_{p} $$
$$(g\circ v)= \int_{0}^{t} g(t-\tau)\bigl\Vert v(t)-v(\tau)\bigr\Vert ^{2}_{2}\,d\tau. $$
Lemma 2.3
[18], Lemma 2.2
Assume (G1) and (G2) hold. Let
\(u(t)\)
be a solution of (1.1). Assume further that
and
where
\(B_{1}=\frac{B}{\sqrt{l}}\). Then there exists a constant
\(\beta >B^{-{\frac{p}{p-2}}}_{1}\)
such that for
\(t>0\)
and
$$E(0)< E_{1}=\biggl(\frac{1}{2}-\frac{1}{p} \biggr)B^{-{\frac{2p}{p-2}}}_{1} $$
$$\Vert \nabla u_{0}\Vert _{2}\geq B^{-{\frac{p}{p-2}}}_{1}, $$
$$ \biggl(1- \int_{0}^{t} g(s)\,ds\biggr)\bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2}\geq\beta^{2} $$
(2.6)
$$ \Vert u\Vert _{p}\geq B_{1}\beta. $$
(2.7)
Lemma 2.4
For
\(2\leq p\leq\rho+2\), we have
$$ \Vert u_{t}\Vert _{p}^{p} \leq \Vert u_{t}\Vert _{p}^{2}+\Vert u_{t}\Vert _{\rho+2}^{\rho+2}. $$
(2.8)
Proof
If \(\Vert u_{t}\Vert _{p}<1\), then we get \(\Vert u_{t}\Vert _{p}^{p}\leq \Vert u_{t}\Vert _{p}^{2}\). If \(\Vert u_{t}\Vert _{p}\geq1\), then have
Together with the two cases, we obtain (2.8). □
$$\Vert u_{t}\Vert _{p}^{p}\leq C\Vert u_{t}\Vert _{p}^{\rho+2}\leq C\Vert u_{t} \Vert _{\rho+2}^{\rho+2}. $$
3 Blow-up result
In this section we state and prove the blow-up result.
Theorem 3.1
Assume that (G1) and (G2) hold and
Assume further that
\(p>\rho+2\), and
\((u_{0}, u_{1})\in H^{1}_{0}(\Omega )\times L^{2}(\Omega)\)
is given. Then the solution
\(u(t)\)
of problem (1.1) blows up in finite time, i.e. there exists
\(T_{0}<+\infty\)
such that
if
and
$$ \int_{0}^{\infty}g(s)\,ds< \frac{p/2-1}{p/2-1+1/2p}. $$
(3.1)
$$ {\lim_{t\rightarrow T_{0}^{-}}} \bigl(\Vert u_{t}\Vert ^{\rho+2}_{\rho+2}+\Vert \nabla u\Vert ^{2}_{2}+ \Vert u\Vert ^{p}_{p} \bigr)=\infty, $$
(3.2)
$$ E(0)< \biggl(1-\frac{1}{p(p-2)}\frac{1-l}{l} \biggr) \biggl( \frac{1}{2}-\frac{1}{p} \biggr)B^{-{\frac{2p}{p-2}}}_{1}, $$
(3.3)
$$ \Vert \nabla u_{0}\Vert _{2}>B^{-{\frac{p}{p-2}}}_{1}. $$
(3.4)
Proof
Assume that there exists some positive constant C such that for \(t>0\) the solution \(u(t)\) of (1.1) satisfies
We set
where the constant \(E_{2}\in(E(0), E_{1})\) shall be chosen later. By Lemma 2.2,
Then, for \(0\leq s\leq t\), we have
From (2.6), we have
Define
where the constant \(\varepsilon>0\) shall be chosen later and the constant σ satisfies
Taking a derivative of (3.9) and using Lemma 2.2, we have
For the last term on the right side of (3.11), using the Green formula, we get
Substituting (3.12) into (3.11), we obtain
Using the Cauchy inequality, for \(0<\varepsilon_{1}<1\) we have
By (3.14), we know
For the fourth term on the right side of (3.15), by (2.2) and Lemma 2.3, we obtain
Then, by (3.15) and (3.16), we have
Since
we have
It is easy to see that there exists \(\varepsilon^{*}_{1}>0\), such that, for \(0<\varepsilon_{1}<\varepsilon^{*}_{1}\),
Since
we may choose \(0<\varepsilon_{1}<1\) sufficiently small, and \(E_{2}\in (E(0),E_{1})\) sufficiently near E(0), such that
Then, for \(t>0\), by (3.17) and (3.19), we obtain
From the above estimate we know there exists a constant \(\gamma>0\) such that
where
Since
combining (3.21), we have
We now estimate the term \(\int_{\Omega} \vert u_{t}\vert ^{\rho }u_{t}u \,dx\) as follows:
Using Young’s inequality, we have
where \(\frac{1}{\mu}+\frac{1}{\theta}=1\). By choosing
we have
By (3.10), we know
Then, by (3.8), we have
where \(k=p-\frac{\theta}{1-\sigma}\) is a positive constant. Now, from (3.23), we have
Therefore it follows
From (3.5) and (3.7), we have
It follows from (3.24) and (3.25) that
Combining (3.21) and (3.26), we arrive at
By a simple integration of (3.27) over \((0, t)\), we obtain
This shows that \(L(t)\) blows up in finite time \(T_{0}\), and
Furthermore, we have
This leads to a contradiction with (3.5). Thus, the solution of problem (1.1) blows up in finite time. □
$$ \Vert u_{t}\Vert ^{\rho+2}_{\rho+2}+ \Vert \nabla u\Vert ^{2}_{2}+\Vert u\Vert ^{p}_{p}\leq C. $$
(3.5)
$$H(t)=E_{2}-E(t), $$
$$ H'(t)=-E'(t)\geq0. $$
(3.6)
$$ 0< H(0)\leq H(s)\leq H(t)=E_{2}-E(t). $$
(3.7)
$$\begin{aligned} H(t) =& E_{2}-E(t) \\ =& E_{2}-\frac{1}{\rho+2}\Vert u_{t}\Vert ^{\rho+2}_{\rho+2}-\frac{1}{2}\biggl(1- \int_{0}^{t} g(s)\,ds\biggr)\bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2}-\frac{1}{2}(g\circ\nabla u) (t)+ \frac{1}{p}\bigl\Vert u(t)\bigr\Vert ^{p}_{p} \\ \leq& E_{2}-\frac{1}{2}\biggl(1- \int_{0}^{t} g(s)\,ds\biggr)\bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2}+\frac{1}{p}\bigl\Vert u(t)\bigr\Vert ^{p}_{p} \\ \leq& E_{1}-\frac{1}{2}\biggl(1- \int_{0}^{t} g(s)\,ds\biggr)\bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2}+\frac{1}{p}\bigl\Vert u(t)\bigr\Vert ^{p}_{p} \\ \leq& E_{1}-\frac{1}{2}B^{-{\frac{2p}{p-2}}}_{1}+ \frac{1}{p}\bigl\Vert u(t)\bigr\Vert ^{p}_{p} \\ \leq& \biggl(\frac{1}{2}-\frac{1}{p}\biggr)B^{-{\frac{2p}{p-2}}}_{1}- \frac{1}{2}B^{-{\frac{2p}{p-2}}}_{1}+\frac{1}{p}\bigl\Vert u(t)\bigr\Vert ^{p}_{p} \\ \leq& \frac{1}{p}\bigl\Vert u(t)\bigr\Vert ^{p}_{p}. \end{aligned}$$
(3.8)
$$ L(t)=H^{1-\sigma}(t)+\frac{\varepsilon}{\rho+1} \int_{\Omega} \vert u_{t}\vert ^{\rho}u_{t}u \,dx+\frac{\varepsilon}{2} \int_{\Omega} \vert \nabla u\vert ^{2}\,dx, $$
(3.9)
$$ 0< \sigma< \frac{1}{\rho+2}-\frac{1}{p}. $$
(3.10)
$$\begin{aligned} L'(t) =& (1-\sigma)H^{-\sigma}(t) \biggl(\bigl\Vert \nabla u_{t}(t)\bigr\Vert ^{2}_{2}- \frac{1}{2}\bigl(g'\circ\nabla u\bigr) (t)+ \frac{1}{2}g(t)\bigl\Vert \nabla u(t)\bigr\Vert ^{2}_{2} \biggr) \\ & {}+\frac{\varepsilon}{\rho+1}\Vert u_{t}\Vert ^{\rho+2}_{\rho+2}+ \varepsilon \int_{\Omega} \vert u_{t}\vert ^{\rho}u_{tt}u \,dx+\varepsilon \int_{\Omega}\nabla u_{t}\nabla u \,dx \\ \geq& (1-\sigma) H^{-\sigma}(t)\bigl\Vert \nabla u_{t}(t) \bigr\Vert ^{2}_{2}+\frac{\varepsilon}{\rho+1}\Vert u\Vert ^{\rho+2}_{\rho+2}-\varepsilon \Vert \nabla u\Vert ^{2}_{2}+\varepsilon \Vert u\Vert ^{p}_{p} \\ &{} -\varepsilon \int_{\Omega} \int_{0}^{t} g(t-\tau)\triangle u(\tau)\,d\tau u(t)\,dx. \end{aligned}$$
(3.11)
$$\begin{aligned}& - \int_{\Omega} \int_{0}^{t} g(t-\tau)\triangle u(\tau)\,d\tau u(t)\,dx \\& \quad= \int_{0}^{t} g(t-\tau) \int_{\Omega}\nabla u(\tau)\nabla u(t)\,dx \,d\tau \\& \quad= \int_{0}^{t} g(t-\tau) \int_{\Omega}\nabla u(t)\nabla\bigl(u(\tau)-u(t)\bigr)\,dx \,d\tau+ \int_{0}^{t} g(t-\tau)\bigl\Vert \nabla u(t)\bigr\Vert ^{2}_{2}\,d\tau \\& \quad= \int_{0}^{t} g(t-\tau) \int_{\Omega}\nabla u(t) \bigl(\nabla u(\tau)-\nabla u(t)\bigr)\,dx\, d \tau+ \int_{0}^{t} g(\tau)\,d\tau\bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2}. \end{aligned}$$
(3.12)
$$\begin{aligned} L'(t) \geq& (1-\sigma) H^{-\sigma}(t)\bigl\Vert \nabla u_{t}(t)\bigr\Vert ^{2}_{2}+\frac{\varepsilon}{\rho+1} \Vert u_{t}\Vert ^{\rho+2}_{\rho+2}+\varepsilon \Vert u\Vert ^{p}_{p} \\ &{} +\varepsilon \int_{0}^{t} g(t-\tau) \int_{\Omega}\nabla u(t) \bigl(\nabla u(\tau)-\nabla u(t)\bigr)\,dx\, d \tau \\ &{} -\varepsilon\biggl(1- \int_{0}^{t} g(\tau)\,d\tau\biggr)\bigl\Vert \nabla u(t)\bigr\Vert ^{2}_{2}. \end{aligned}$$
(3.13)
$$\begin{aligned}& \int_{0}^{t} g(t-\tau) \int_{\Omega}\nabla u(t) \bigl(\nabla u(\tau)-\nabla u(t)\bigr)\,dx\, d \tau \\& \quad\geq-\frac{p(1-\varepsilon_{1})}{2} \int_{0}^{t} g(t-\tau)\bigl\Vert \nabla u(\tau)- \nabla u(t)\bigr\Vert ^{2}_{2}\,d\tau \\& \qquad{} -\frac{1}{(1-\varepsilon_{1})2p} \int_{0}^{t} g(\tau)\,d\tau\bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2}. \end{aligned}$$
(3.14)
$$\begin{aligned} L'(t) \geq& (1-\sigma) H^{-\sigma}(t)\bigl\Vert \nabla u_{t}(t)\bigr\Vert ^{2}_{2}+ \frac{\varepsilon}{\rho+1}\Vert u_{t}\Vert ^{\rho+2}_{\rho+2}+ \varepsilon \Vert u\Vert ^{p}_{p} \\ &{}-\varepsilon\biggl(1- \int_{0}^{t} g(\tau)\,d\tau\biggr)\bigl\Vert \nabla u(t)\bigr\Vert ^{2}_{2} \\ & {}-\varepsilon\biggl(\frac{p(1-\varepsilon_{1})}{2} \int_{0}^{t} g(t-\tau)\bigl\Vert \nabla u(\tau)- \nabla u(t)\bigr\Vert ^{2}_{2}\,d\tau \\ &{}+\frac{1}{(1-\varepsilon_{1})2p} \int_{0}^{t} g(\tau)\,d\tau\bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2} \biggr) \\ \geq& (1-\sigma) H^{-\sigma}(t)\bigl\Vert \nabla u_{t}(t) \bigr\Vert ^{2}_{2}+\varepsilon\biggl(\frac{1}{\rho+1}+ \frac{p(1-\varepsilon_{1})}{\rho+2} \biggr)\Vert u_{t}\Vert ^{\rho +2}_{\rho+2}+ \varepsilon(1-\varepsilon_{1})p\bigl(E_{2}-E(t)\bigr) \\ & {}+\varepsilon\biggl( \biggl(\frac{p(1-\varepsilon_{1})}{2}-1 \biggr) \biggl(1- \int_{0}^{t} g(\tau)\,d\tau\biggr)\bigl\Vert \nabla u(t)\bigr\Vert ^{2}_{2} \\ &{}-\frac{1}{(1-\varepsilon_{1})2p} \int_{0}^{t} g(\tau)\,d\tau\bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2} \biggr) \\ &{} -\varepsilon(1-\varepsilon_{1})pE_{2}+\varepsilon \varepsilon_{1}\Vert u\Vert ^{p}_{p} \\ =& (1-\sigma) H^{-\sigma}(t)\bigl\Vert \nabla u_{t}(t)\bigr\Vert ^{2}_{2}+\varepsilon\biggl(\frac{1}{\rho+1}+ \frac{p(1-\varepsilon_{1})}{\rho+2} \biggr)\Vert u_{t}\Vert ^{\rho +2}_{\rho+2}+ \varepsilon(1-\varepsilon_{1})pH(t) \\ & {}+\varepsilon\biggl( \biggl(\frac{p(1-\varepsilon_{1})}{2}-1 \biggr) \biggl(1- \int_{0}^{t} g(\tau)\,d\tau\biggr)\bigl\Vert \nabla u(t)\bigr\Vert ^{2}_{2} \\ &{}-\frac{1}{(1-\varepsilon_{1})2p} \int_{0}^{t} g(\tau)\,d\tau\bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2} \biggr) \\ &{} -\varepsilon(1-\varepsilon_{1})pE_{2}+\varepsilon \varepsilon_{1}\Vert u\Vert ^{p}_{p}. \end{aligned}$$
(3.15)
$$\begin{aligned}& \biggl(\frac{p(1-\varepsilon_{1})}{2}-1 \biggr) \biggl(1- \int_{0}^{t} g(\tau)\,d\tau\biggr)\bigl\Vert \nabla u(t)\bigr\Vert ^{2}_{2}-\frac{1}{(1-\varepsilon_{1})2p} \int_{0}^{t} g(\tau)\,d\tau\bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2} \\& \quad= \frac{ (\frac{p(1-\varepsilon_{1})}{2}-1) (1-\int_{0}^{t} g(\tau)\, d\tau)-\frac{1}{(1-\varepsilon_{1})2p}\int_{0}^{t} g(\tau)\,d\tau}{1-\int _{0}^{t} g(\tau)\,d\tau} \biggl(1- \int_{0}^{t} g(\tau)\,d\tau\biggr)\bigl\Vert \nabla u(t)\bigr\Vert ^{2}_{2} \\& \quad\geq\frac{ (\frac{p(1-\varepsilon_{1})}{2}-1)l -\frac{1}{(1-\varepsilon_{1})2p}(1-l)}{1-\int_{0}^{t} g(\tau)\,d\tau}\beta^{2}. \end{aligned}$$
(3.16)
$$\begin{aligned} L'(t) \geq& (1-\sigma) H^{-\sigma}(t)\bigl\Vert \nabla u_{t}(t)\bigr\Vert ^{2}_{2}+ \varepsilon\biggl(\frac{1}{\rho+1}+\frac{p(1-\varepsilon_{1})}{\rho+2} \biggr)\Vert u_{t}\Vert ^{\rho+2}_{\rho+2}+ \varepsilon(1- \varepsilon_{1})pH(t) \\ & {}+\frac{ (\frac{p(1-\varepsilon_{1})}{2}-1)l -\frac{1}{(1-\varepsilon_{1})2p}(1-l)}{1-\int_{0}^{t} g(\tau)\,d\tau }\varepsilon\beta^{2}-\varepsilon(1- \varepsilon_{1})pE_{2}+\varepsilon\varepsilon_{1} \Vert u\Vert ^{p}_{p}. \end{aligned}$$
(3.17)
$$\begin{aligned} \int_{0}^{\infty}g(s)\,ds< \frac{p/2-1}{p/2-1+1/2p}, \end{aligned}$$
$$\begin{aligned} \biggl(\frac{p}{2}-1 \biggr) \biggl(1- \int_{0}^{\infty}g(\tau)\,d\tau\biggr)- \frac{1}{2p} \int_{0}^{\infty}g(\tau)\,d\tau>0. \end{aligned}$$
$$\begin{aligned} \frac{ (\frac{p(1-\varepsilon_{1})}{2}-1)l-\frac{1}{(1-\varepsilon _{1})2p}(1-l)}{1-\int_{0}^{t} g(\tau)\,d\tau}\beta^{2}>\frac{ (\frac {p(1-\varepsilon_{1})}{2}-1)l-\frac{1}{(1-\varepsilon _{1})2p}(1-l)}{1-\int_{0}^{t} g(\tau)\,d\tau}B^{-\frac{2p}{p-2}}_{1}. \end{aligned}$$
(3.18)
$$\begin{aligned} E(0)< \biggl(\frac{1}{2}-\frac{1}{p} \biggr) \biggl(1- \frac{1}{p(p-2)}\frac{1-l}{l} \biggr)B^{-\frac{2p}{p-2}}_{1}= \frac{(\frac{p}{2}-1)l- \frac{1}{2p}(1-l)}{pl}B^{-\frac{2p}{p-2}}_{1}, \end{aligned}$$
$$\begin{aligned} \frac{(\frac{p(1-\varepsilon_{1})}{2}-1)l-\frac {1}{(1-\varepsilon_{1})2p}(1-l)}{1-\int_{0}^{t} g(\tau)\,d\tau}B^{-\frac {2p}{p-2}}_{1}-p(1- \varepsilon_{1})E_{2}\geq0. \end{aligned}$$
(3.19)
$$\begin{aligned} L'(t) \geq& (1-\sigma) H^{-\sigma}(t)\bigl\Vert \nabla u_{t}(t)\bigr\Vert ^{2}_{2}+\varepsilon\biggl( \frac{1}{\rho+1}+\frac{p(1-\varepsilon_{1})}{\rho+2} \biggr)\Vert u_{t}\Vert ^{\rho+2}_{\rho+2} \\ & {}+\varepsilon(1-\varepsilon_{1})pH(t)+\varepsilon \varepsilon_{1}\Vert u\Vert ^{p}_{p}. \end{aligned}$$
(3.20)
$$\begin{aligned} L'(t)\geq\varepsilon\gamma\bigl(\Vert u_{t}\Vert ^{\rho+2}_{\rho+2}+H(t)+\Vert u\Vert ^{p}_{p} \bigr)\geq0,\quad t>0, \end{aligned}$$
(3.21)
$$\begin{aligned} \gamma=\mbox{min} \biggl\{ \biggl(\frac{1}{\rho+1}+\frac{p(1-\varepsilon _{1})}{\rho+2} \biggr), (1-\varepsilon_{1})p, \varepsilon_{1} \biggr\} . \end{aligned}$$
$$\begin{aligned} L(0)=H^{1-\sigma}(0)+\frac{\varepsilon}{\rho+1} \int_{\Omega} \vert u_{t}\vert ^{\rho}u_{1}u_{0}\,dx+\frac {\varepsilon}{2} \int_{\Omega} \vert \nabla u_{0}\vert ^{2}\,dx>0, \end{aligned}$$
$$\begin{aligned} L(t)\geq L(0)>0,\quad t>0. \end{aligned}$$
$$\begin{aligned} \biggl\vert \int_{\Omega} \vert u_{t}\vert ^{\rho}u_{t}u \,dx\biggr\vert \leq \Vert u_{t}\Vert ^{\rho+1}_{\rho+2} \Vert u\Vert _{\rho+2}\leq C \Vert u_{t}\Vert ^{\rho+1}_{\rho+2}\Vert u\Vert _{p}. \end{aligned}$$
$$\begin{aligned} \biggl(\biggl\vert \int_{\Omega} \vert u_{t}\vert ^{\rho}u_{t}u \,dx\biggr\vert \biggr)^{\frac{1}{1-\sigma}}\leq C\Vert u_{t}\Vert ^{\frac{\rho+1}{1-\sigma}}_{\rho+2}\Vert u\Vert ^{\frac{1}{1-\sigma }}_{p}\leq C \bigl(\Vert u_{t}\Vert ^{\frac{\rho+1}{1-\sigma}\mu}_{\rho +2}+\Vert u \Vert ^{\frac{1}{1-\sigma}\theta}_{p} \bigr), \end{aligned}$$
$$\begin{aligned} \mu=\frac{(1-\sigma)(\rho+2)}{\rho+1}>1, \end{aligned}$$
$$\begin{aligned} \frac{\theta}{1-\sigma}=\frac{\rho+2}{(1-\sigma)(\rho+2)-(\rho+1)}. \end{aligned}$$
$$\begin{aligned} \frac{\theta}{1-\sigma}< p. \end{aligned}$$
(3.22)
$$\begin{aligned} \Vert u\Vert ^{\frac{\theta}{1-\sigma}}_{p}=\Vert u\Vert ^{p-(p-\frac{\theta}{1-\sigma})}_{p}=\Vert u\Vert ^{p}_{p} \Vert u\Vert ^{-k}_{p}\leq \Vert u\Vert ^{p}_{p}CH^{-\frac{k}{p}}(t)\leq C\Vert u\Vert ^{p}_{p}H^{-\frac{k}{p}}(0), \end{aligned}$$
(3.23)
$$\begin{aligned} \biggl(\biggl\vert \int_{\Omega} \vert u_{t}\vert ^{\rho}u_{t}u \,dx\biggr\vert \biggr)^{\frac{1}{1-\sigma}}\leq C\bigl(\Vert u_{t}\Vert ^{\rho+2}_{\rho+2}+\Vert u\Vert ^{p}_{p}H^{-\frac{p}{k}}(0) \bigr). \end{aligned}$$
$$\begin{aligned} L^{\frac{1}{1-\sigma}}(t) =& \biggl(H^{1-\sigma}(t)+ \frac{\varepsilon}{\rho+1} \int_{\Omega} \vert u_{t}\vert ^{\rho}u_{t}u \,dx+\frac{\varepsilon}{2} \int_{\Omega}\bigl\vert \nabla u(t)\bigr\vert ^{2}\,dx \biggr)^{\frac{1}{1-\sigma}} \\ \leq& C \biggl( H(t)+\biggl\vert \int_{\Omega} \vert u_{t}\vert ^{\rho}u_{t}u \,dx\biggr\vert ^{\frac{1}{1-\sigma}}+\biggl\vert \int_{\Omega}\bigl\vert \nabla u(t)\bigr\vert ^{2}\,dx \biggr\vert ^{\frac{1}{1-\sigma}} \biggr) \\ \leq& C \bigl(H(t)+\Vert u_{t}\Vert ^{\rho+2}_{\rho+2}+ \Vert u\Vert ^{p}_{p}+\bigl\Vert \nabla u(t)\bigr\Vert ^{\frac{2}{1-\sigma}}_{2} \bigr). \end{aligned}$$
(3.24)
$$\begin{aligned} \bigl\Vert \nabla u(t)\bigr\Vert ^{\frac{2}{1-\sigma}}_{2} \leq C^{\frac{1}{1-\sigma}}\leq\frac{C^{\frac{1}{1-\sigma}}}{H(0)}H(t). \end{aligned}$$
(3.25)
$$\begin{aligned} L^{\frac{1}{1-\sigma}}(t)\leq C \bigl(H(t)+\Vert u_{t} \Vert ^{\rho+2}_{\rho+2}+\Vert u\Vert ^{p}_{p} \bigr). \end{aligned}$$
(3.26)
$$\begin{aligned} L'(t)>\frac{\varepsilon\gamma}{C}L^{\frac{1}{1-\sigma }}(t),\quad t>0. \end{aligned}$$
(3.27)
$$\begin{aligned} L^{\sigma/(1-\sigma)}(t)\geq\frac{1}{L^{-\sigma/(1-\sigma )}(0)-\varepsilon\gamma t\sigma/[C(1-\sigma)]},\quad t>0. \end{aligned}$$
$$\begin{aligned} T_{0}\leq\frac{C(1-\sigma)}{\varepsilon\gamma\sigma L^{\sigma/(1-\sigma)}(0)}. \end{aligned}$$
$$\begin{aligned} {\lim_{t\rightarrow T_{0}^{-}}} \bigl(\Vert u_{t}\Vert ^{\rho+2}_{\rho+2}+\Vert \nabla u\Vert ^{2}_{2}+ \Vert u\Vert ^{p}_{p} \bigr)=\infty. \end{aligned}$$
(3.28)
4 Global existence
In this section we show that the solution of (1.1) is global if \(\rho+2\geq p\).
Theorem 4.1
Assume that (G1), (G2) hold and
\(2< p\leq\rho+2\). Assume further
Then for any initial data
\((u_{0}, u_{1})\in H_{0}^{1}(\Omega)\times L^{2}(\Omega)\), the solution of problem (1.1) exists globally.
$$\rho+2\leq\frac{2(n-1)}{n-2}. $$
Proof
We set
Differentiating \(F(t)\) and using (2.5), we get
Using the Hölder inequality and Young’s inequality, we obtain the estimate
in which ε̃ is a small positive constant to be chosen later, and \(C(\tilde{\varepsilon})\) is a positive constant depending on ε̃.
$$\begin{aligned} F(t) =& \frac{1}{\rho+2}\Vert u_{t}\Vert ^{\rho+2}_{\rho+2}+ \frac{1}{2}\biggl(1 - \int_{0}^{t} g(s)\,ds\biggr)\bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2}+ \frac{1}{2}(g\circ\nabla u) (t) + \frac{1}{p}\bigl\Vert u(t)\bigr\Vert ^{p}_{p} \\ =& E(t)+\frac{2}{p}\bigl\Vert u(t)\bigr\Vert ^{p}_{p}. \end{aligned}$$
(4.1)
$$\begin{aligned} F'(t)=E'(t)+2 \int_{\Omega} \vert u\vert ^{p-2}uu_{t}\,dx\leq -\Vert \nabla{u}_{t}\Vert ^{2}_{2}+2 \int_{\Omega} \vert u\vert ^{p-2}uu_{t}\,dx. \end{aligned}$$
(4.2)
$$\begin{aligned} \biggl\vert \int_{\Omega} \vert u\vert ^{p-2}uu_{t}\,dx \biggr\vert \leq\tilde{\varepsilon} \Vert u_{t}\Vert ^{p}_{p}+C(\tilde{\varepsilon})\Vert u\Vert ^{p}_{p}, \end{aligned}$$
(4.3)
Using Lemma 2.4, (4.3) and the embedding theorem, we obtain
Substituting (4.4) to (4.2), we have
Choosing \(\tilde{\varepsilon}={1\over 2B}\) in (4.5), we arrive at
Furthermore we obtain
This completes the proof of the global existence result. □
$$\begin{aligned} \biggl\vert \int_{\Omega} \vert u\vert ^{p-2}uu_{t}\,dx \biggr\vert \leq& \tilde{\varepsilon} \Vert u_{t}\Vert ^{2}_{p}+C_{1}\tilde{\varepsilon} \Vert u_{t}\Vert ^{\rho+2}_{\rho+2} +C(\tilde{\varepsilon}) \Vert u\Vert ^{p}_{p} \\ \leq& \tilde{\varepsilon}B \Vert \nabla u_{t}\Vert ^{2}_{2}+C_{1}\tilde{\varepsilon} \Vert u_{t}\Vert ^{\rho+2}_{\rho+2} +C(\tilde{\varepsilon}) \Vert u\Vert ^{p}_{p}. \end{aligned}$$
(4.4)
$$\begin{aligned} F'(t)\leq-(1-\tilde{\varepsilon}B)\Vert \nabla u_{t}\Vert ^{2}_{2}+C_{1}\tilde{ \varepsilon} \Vert u_{t}\Vert ^{\rho+2}_{\rho+2}+C( \tilde{\varepsilon})\Vert u\Vert ^{p}_{p}. \end{aligned}$$
(4.5)
$$F'(t)\leq C_{2}F(t). $$
$$F(t)\leq C_{3}e^{C_{2}t}. $$
Declarations
Acknowledgements
The authors are highly grateful for the referees valuable suggestions which improved this work a lot. This work was supported by NNSF of China (61374089), NSF of Shanxi Province (2014011005-2), Shanxi international science and technology cooperation projects (2014081026).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
(1)
School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi, 030006, China
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