Research
Open Access
Blow-up of solution for an integro-differential equation with arbitrary positive initial energy
- Liu Jie1 and
- Liang Fei1Email author
Received: 29 January 2015
Accepted: 28 May 2015
Published: 12 June 2015
Abstract
In this paper, we consider the integro-differential equation \(u_{tt}-M(\|\nabla u\|^{2}_{2})\Delta u+ \int_{0}^{t} g(t-\tau)\Delta u(\tau)\, d\tau+ u_{t}=f(u)\), \((x,t)\in\Omega\times(0,T)\), with initial and Dirichlet boundary conditions. Under suitable assumptions on the functions g and the initial data, a blow-up result with arbitrary positive initial energy is established.
Keywords
blow-up arbitrary positive initial energy integro-differential equationMSC
35L05 35L55 35L701 Introduction
In this paper we study the following nonlinear integro-differential equations: where Ω is a bounded domain in \(\mathbb{R}^{n}\) with a smooth boundary ∂Ω, M is a positive \(C^{1}\)-function like \(M(s)=m_{0}+bs^{\gamma}\), \(m_{0}>0\), \(b\geq0\), \(\gamma\geq1\), and \(s\geq0\), g represents the kernel of the memory term and f is a nonlinear function like \(f(u)=|u|^{p-2}u\), \(p>2\), they will be specified later.
$$ \textstyle\begin{cases} u_{tt}-M(\|\nabla u\|^{2}_{2})\Delta u+ \int_{0}^{t} g(t-\tau)\Delta u(\tau)\, d\tau+u_{t}=f(u), &(x,t)\in\Omega\times (0,T), \\ u(x,t)=0, &x\in\partial\Omega\times(0,T), \\ u(x,0)=u_{0}(x), \qquad u_{t}(x,0)=u_{1}(x), &x\in\Omega, \end{cases} $$
(1.1)
Before going further, (1.1) without the viscoelastic term, that is, \(g\equiv0\), for the case that \(M\equiv1\), (1.1) becomes a nonlinear wave equation which has been extensively studied and several results concerning existence and nonexistence have been established [1–5]. When M is not a constant function, a special case of (1.1) is Kirchhoff equation which has been introduced in order to describe the nonlinear vibrations of an elastic string. Kirchhoff [6] was the first one to study the oscillations of stretched strings and plates. In this case the existence and nonexistence of solutions have been discussed by many authors; see [7–11] and the references cited therein.
For (1.1) with \(g \neq0\), in the case that \(M \equiv 1\), (1.1) becomes a semilinear viscoelastic equation which has been extensively studied and many results concerning global existence and blow-up in finite time have been proved. See in this regard [12–16]. For instance, Messaoudi [13] studied (1.1) with damping term \(a|u_{t}|^{m-2}u_{t}\) and \(f(u)=b|u|^{p-2}u\) and proved a blow-up result for solutions with negative initial energy if \(p>m\geq2\) and a global result for \(2\leq p\leq m\). This result has later been improved by the same author in [14] to accommodate certain solutions with positive initial energy. In [15], Song and Zhong considered (1.1) with strong damping \(-\Delta u_{t}\) and \(f(u)=|u|^{p-2}u\) and proved a blow-up result for solutions with positive initial energy by using the ideas of the ‘potential well’ theory introduced by Payne and Sattinger [5].
For \(g\neq0\) and M is not a constant function, (1.1) is a model to describe the motion of deformable solids as hereditary effect is incorporated. It may also be used to describe the dynamics of an extensible string with fading memory. This equation states that the dynamic equilibrium of a body depends not only on the present state of deformation, but also on the previous history of the deformation [17]. Also, (1.1) is applied to the theory of the heat conduction with memory; see [18, 19]. Therefore, the dynamics of (1.1) is of great importance and interest as they have wide applications in natural sciences.
This type of problem have been considered by many authors and several results concerning existence, nonexistence, and asymptotic behavior have been established. Equation (1.1) was first studied by Torrejón and Young [20], who proved the existence of weakly asymptotic stable solution for a large analytical datum. Later, Munoz Rivera [17] showed the existence of global solutions for small datum and the total energy decays to zero exponentially under some restrictions. In [11], Wu and Tsai studied (1.1) for a strong damping \(-\Delta u_{t}\) and proved the global existence, decay result, and blow-up properties. Recently, they [21] discussed the local existence and blow-up of solutions with positive initial energy for nonlinear damping under some conditions.
In this paper, we consider problem (1.1) and will establish a blow-up result for (1.1) with arbitrary positive initial energy under suitable assumptions on the functions g and the initial data. This result extends earlier ones [11, 21], in which only some a positive initial energy is considered. The main tool in proving blow-up result is the ‘concavity method’ where the basic idea of the method is to construct a positive defined functional \(F(t)\) of the solution by the energy inequality and show that \(F^{-\alpha}(t)\) is a concave function of t.
The rest of this paper is organized as follows. In Section 2, we give some preliminaries and state the main result. In Section 3, we prove the blow-up result by a concavity method. Section 4 is devoted to a simple discussion of the main result.
2 Preliminaries
First, let us introduce some notation used throughout this paper. We denote by \(\|\cdot\|_{q}\) the \(L^{q}(\Omega)\) norm for \(1\leq q\leq\infty\) and by \(\|\nabla\cdot\|_{2}\) the Dirichlet norm in \(H^{1}_{0}(\Omega)\), which is equivalent to the \(H^{1}(\Omega)\) norm. Moreover, we set as the usual \(L^{2}(\Omega)\) inner product.
$$(\varphi,\psi)=\int_{\Omega}\varphi(x)\psi(x)\, dx $$
We next state some assumptions on f, M, and g:
- (A1)\(f(0)=0\) and there are two positive constants \(c_{1}\) and δ such thatand$$\bigl\vert f(s)-f \bigl(s' \bigr) \bigr\vert \leq c_{1} \bigl\vert s-s' \bigr\vert \bigl(|s|^{p-2}+ \bigl\vert s' \bigr\vert ^{p-2} \bigr) $$for \(s, s'\in\mathcal{R}\), \(2< p\leq\frac{2(n-1)}{n-2}\) if \(n>2\) and \(2< p<\infty\) if \(n\leq2\), where \(F(s)=\int^{s}_{0} f(\tau)\, d\tau\).$$sf(s)\geq(2+4\delta)F(s) $$
- (A2)M is a positive \(C^{1}\)-function like \(M(s)=m_{0}+bs^{\gamma}\), \(m_{0}>0\), \(b\geq0\), \(\gamma\geq1\), and \(s\geq0\), and it satisfieswhere \(\overline{M}(s)=\int_{0}^{s} M(\tau)\, d\tau\) and δ is the constant appeared in (A1).$$(2\delta+1)\overline{M}(s)- \bigl(M(s)+2\delta m_{0} \bigr)s\geq0, \quad \forall s\geq0, $$
- (A3)Assume \(g(t):\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}\) belongs to \(C^{1}(\mathbb{R}_{+})\) and satisfyand$$g(t)\geq0, \qquad g'(t)\leq0\quad \mbox{for } t\geq0 $$$$l=\int_{0}^{\infty}g(s)\, ds< \frac{2\delta m_{0}}{1+2\delta}. $$
- (A4)The function \(e^{\frac{t}{2}}g(t)\) is of positive type in the following sense:\(\forall\upsilon\in C^{1} ([0,\infty) )\) and \(\forall t>0\).$$\int_{0}^{t} \upsilon(s)\int_{0}^{s} e^{\frac{s-\tau}{2}}g(s-\tau)\, d\tau\, ds\geq0, $$
Remark 2.1
It is clear that \(f(u)=|u|^{p-2}u\), \(p\geq2\gamma+2\), and \(m(s)=m_{0}+bs^{\gamma}\), where \(m_{0}>0\), \(b\geq0\), \(\gamma\geq1\) satisfy the assumptions (A1) and (A2) with \(\alpha/2\leq\delta\leq(p-2)/4\). It is also obvious that \(g(t)=\epsilon e^{-t}\) with \(0<\epsilon<m_{0}\) satisfies the assumptions (A3) and (A4).
Now we are ready to state the local existence of problem (1.1), whose proof can be found in [21].
Theorem 2.1
Assume that (A1)-(A3) hold, and that
\(u_{0}\in H_{0}^{1}(\Omega)\cap H^{2}(\Omega)\), \(u_{1}\in L^{2}(\Omega)\), then there exists a unique solution
u
of (1.1) satisfying
Moreover, at least one of the following statements holds true:
$$u\in C \bigl( [0,T );H^{1}_{0}(\Omega)\cap H^{2}(\Omega) \bigr) \quad \textit{and}\quad u_{t}\in C \bigl( [0,T );L^{2}(\Omega) \bigr)\cap L^{2} \bigl( [0,T );H_{0}^{1}(\Omega) \bigr). $$
- (i)
\(T=\infty\),
- (ii)
\(\|u_{t}(t)\|_{2}^{2}+\|\Delta u(t)\|_{2}^{2}\rightarrow\infty\) as \(t\rightarrow T^{-}\).
We introduce the energy functional \(E(t)\) associated to our equation, where As in [21], we see that which implies
$$\begin{aligned} E(t) =&\frac{1}{2} \bigl\Vert u_{t}(t) \bigr\Vert ^{2}_{2}+\frac{1}{2}\overline{M} \bigl( \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2} \bigr)- \frac{1}{2}\int_{0}^{t} g(\tau)\, d\tau \bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2} \\ &{}+\frac{1}{2}(g \circ\nabla u) (t)- \int_{\Omega}F(u)\, dx, \end{aligned}$$
(2.1)
$$(g\circ\nabla w) (t)=\int_{0}^{t} g(t-\tau) \bigl\Vert \nabla w(t)-\nabla w(\tau ) \bigr\Vert ^{2}_{2} \, d \tau. $$
$$\frac{d}{dt}E(t)=-\int_{\Omega}\bigl\vert u_{t}(t) \bigr\vert ^{2}\, dx+\frac{1}{2} \bigl(g'\circ \nabla u \bigr) (t)-\frac{1}{2}g(t) \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2}\leq0, $$
$$ E(t)\leq E(0)-\int_{0}^{t}\int _{\Omega}\bigl\vert u_{t}(\tau) \bigr\vert ^{2}\, dx\, d\tau. $$
(2.2)
Let for \(u\in H^{1}_{0}(\Omega)\). We finally state our main blow-up result for problem (1.1).
$$I(u)=M \bigl( \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2} \bigr) \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2}-\int _{\Omega}f(u)u\, dx $$
Theorem 2.2
Assume (A1)-(A4) hold. If
\(u_{0}\in H_{0}^{1}(\Omega)\cap H^{2}(\Omega)\), \(u_{1}\in L^{2}(\Omega)\), satisfy the following conditions:
and
where
δ
is the constant appeared in (A1) and
λ
is the constant of the Poincaré inequality on Ω, then the corresponding solution
\(u(t)\)
of problem (1.1) blows up in a finite time
\(T^{*}>0\).
$$ E(0)>0, \qquad \int_{\Omega}u_{0}(x)u_{1}(x) \, dx>0 $$
(2.3)
$$ I(u_{0})< 0, \qquad \|u_{0} \|_{2}^{2}>\frac{(4\delta+2)E(0)}{ (2\delta m_{0}-(2\delta+1)l )\min\{\lambda,1\}}, $$
(2.4)
3 Proof of Theorem 2.2
In this section, we deal with the blow-up solutions of equation (1.1). Before we prove our blow-up result, we need the following lemmas.
Lemma 3.1
(see [16], Lemma 2.1)
Assume that
\(g(t)\)
satisfies the assumptions (A3) and (A2), and
\(\Lambda(t)\)
is a function that is twice continuously differentiable, satisfying
for every
\(t\in[0,T_{0})\), where
\(u(t)\)
is the corresponding solution of (1.1) with
\(u_{0}\)
and
\(u_{1}\). Then the function
\(\Lambda(t)\)
is strictly increasing on
\([0,T_{0})\).
$$ \textstyle\begin{cases} \Lambda''(t)+\Lambda'(t)>\int_{0}^{t}g(t-\tau)\int_{\Omega}\nabla u(\tau,x)\nabla u(t,x)\, dx\, d\tau, \\ \Lambda(0)>0,\qquad \Lambda'(0)>0 \end{cases} $$
Lemma 3.2
Assume
\(u_{0}\in H_{0}^{1}(\Omega)\cap H^{2}(\Omega)\), \(u_{1}\in L^{2}(\Omega)\), satisfy
If the local solution
\(u(t)\)
of (1.1) satisfies
then
\(\|u(t)\|_{2}^{2}\)
is strictly increasing on
\([0,T)\).
$$\int_{\Omega}u_{0}(x)u_{1}(x)\, dx>0. $$
$$I(u)< 0, $$
Proof
Since \(u(t)\) is the local solution of (1.1), by a simple computation we have where the last inequality uses \(I(u)<0\), which implies Therefore, this lemma comes from Lemma 3.1. □
$$\begin{aligned} \frac{1}{2}\frac{d^{2}}{dt^{2}}\int_{\Omega}\bigl\vert u(t,x) \bigr\vert ^{2}\, dx =&\int_{\Omega}\bigl( \bigl\vert u_{t}(t) \bigr\vert ^{2}+uu_{tt} \bigr)\, dx \\ =&\int_{\Omega}\bigl\vert u_{t}(t,x) \bigr\vert ^{2}\, dx-\int_{\Omega}uu_{t}\, dx-M \bigl( \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2} \bigr) \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2} \\ &{}+\int _{\Omega}f(u)u\, dx +\int_{0}^{t}g(t-\tau)\int _{\Omega}\nabla u(t)\cdot\nabla u(\tau)\, dx\, d\tau \\ >&-\int_{\Omega}uu_{t}\, dx+\int _{0}^{t}g(t-\tau)\int_{\Omega}\nabla u(t)\cdot \nabla u(\tau)\, dx\, d\tau, \end{aligned}$$
$$\frac{d^{2}}{dt^{2}}\int_{\Omega}\bigl\vert u(t,x) \bigr\vert ^{2}\, dx+\frac{d}{dt}\int_{\Omega}\bigl\vert u(t,x) \bigr\vert ^{2}\, dx>\int_{0}^{t}g(t- \tau)\int_{\Omega}\nabla u(t)\cdot\nabla u(\tau)\, dx\, d\tau. $$
Proof of Theorem 2.2
We next prove Theorem 2.2 in two steps. First, by a contradiction argument we claim that and for every \(t\in[0,T)\). If this was not the case, then there would exist a time \(t_{1}\) such that By the continuity of the solution \(u(t)\) as a function of t, we see that \(I (u(t) )<0\) when \(t\in(0,t_{1})\) and \(I (u(t_{1}) )=0\). Thus by Lemma 3.1 we have for every \(t\in[0,t_{1})\). In addition, it is obvious that \(\|u(t)\|^{2}_{2}\) is continuous on \([0, t_{1}]\). Thus the following inequality is obtained: On the other hand, it follows from the definition of \(E(t)\) and (2.2) that Using the assumptions (A1) and (A2), we have Noting the fact that \(I (u(t_{1}) )=0\), we then have Thus, by the Poincaré inequality, we have Obviously, there is a contradiction between (3.4) and (3.6). Thus, we have proved that (3.1) is true for every \(t\in[0,T)\). Furthermore, by Lemma 3.2 we see that (3.2) is also valid on \(t\in[0,T)\).
$$ I \bigl(u(t) \bigr)< 0 $$
(3.1)
$$ \bigl\Vert u(t) \bigr\Vert ^{2}_{2}> \frac{(4\delta+2)E(0)}{ (2\delta m_{0}-(2\delta+1)l )\min\{\lambda,1\}} $$
(3.2)
$$ t_{1}=\min \bigl\{ t\in(0,T): I \bigl(u(t) \bigr)=0 \bigr\} >0. $$
(3.3)
$$\bigl\Vert u(t) \bigr\Vert ^{2}_{2}>\|u_{0} \|^{2}_{2}>\frac{(4\delta+2)E(0)}{ (2\delta m_{0}-(2\delta+1)l )\min\{\lambda,1\}} $$
$$ \bigl\Vert u(t_{1}) \bigr\Vert ^{2}_{2}> \frac{(4\delta+2)E(0)}{ (2\delta m_{0}-(2\delta+1)l )\min\{\lambda,1\}}. $$
(3.4)
$$\begin{aligned}& \frac{1}{2}\overline{M} \bigl( \bigl\Vert \nabla u(t_{1}) \bigr\Vert _{2}^{2} \bigr)- \frac{1}{2}\int_{0}^{t_{1}} g(\tau)\, d\tau \bigl\Vert \nabla u(t_{1}) \bigr\Vert ^{2}_{2} \\& \quad {}+ \frac{1}{2}(g\circ\nabla u) (t_{1})-\int_{\Omega}F \bigl(u(t_{1}) \bigr)\, dx\leq E(0). \end{aligned}$$
(3.5)
$$\begin{aligned}& M \bigl( \bigl\Vert \nabla u(t_{1}) \bigr\Vert _{2}^{2} \bigr) \bigl\Vert \nabla u(t_{1}) \bigr\Vert _{2}^{2}+2 \delta m_{0} \bigl\Vert \nabla u(t_{1}) \bigr\Vert _{2}^{2}-(2\delta+1)l \bigl\Vert \nabla u(t_{1}) \bigr\Vert _{2}^{2} \\& \quad {}-\int _{\Omega}f \bigl(u(t_{1}) \bigr)u(t_{1})\, dx \leq(4\delta+2)E(0). \end{aligned}$$
$$\bigl(2\delta m_{0}-(2\delta+1)l \bigr) \bigl\Vert \nabla u(t_{1}) \bigr\Vert ^{2}_{2}\leq(4\delta+2)E(0). $$
$$ \bigl\Vert u(t_{1}) \bigr\Vert ^{2}_{2} \leq\frac{(4\delta+2)E(0)}{ (2\delta m_{0}-(2\delta+1)l )\min\{\lambda,1\}}. $$
(3.6)
Secondly, we prove that the solution of problem (1.1) blows up in a finite time. Assume by contradiction that the solution u is global. Then, for sufficiently large \(T>0\), we consider \(H(t):[0,T]\rightarrow\mathbb{R}_{+}\) defined by where \(t_{0}\) and α are positive constants, which will be determined in the sequel. A direct computation yields and Therefore, we have where \(\Psi(t),G(t):[0,T]\rightarrow\mathbb{R}_{+}\) are the functions defined by and Using the Schwarz inequality, we have and and Similarly, we have The previous inequalities entail \(G(t)\geq0\) for every \([0,T]\). Using (3.7), we get where Using Young’s inequality, we have Inserting (3.10) into (3.9), we have Using the assumptions (A1) and (A2), we have where the last inequality follows from Lemma 3.2 and the Poincaré inequality. From (2.4), we have Thus, we can let α satisfy which implies that there exists \(\theta>0\) (independent of T) such that By (3.8) and (3.11), it follows that Moreover, we let \(t_{0}\) satisfy which means \(H'(0)>0\). Thus by \(H''(t)>0\) we see that \(H(t)\) and \(H'(t)\) are strictly increasing on \([0,T]\).
$$H(t)=\bigl\Vert u(t)\bigr\Vert _{2}^{2}+\int_{0}^{t} \bigl\Vert u(\tau) \bigr\Vert ^{2}_{2}\, d\tau +(T-t) \|u_{0}\|_{2}^{2}+\alpha(t_{0}+t)^{2}, $$
$$\begin{aligned} H'(t)&=2\int_{\Omega}u(t)u_{t}(t)\, dx+ \bigl\Vert u(t) \bigr\Vert ^{2}_{2}-\|u_{0} \|_{2}^{2}+2\alpha (t_{0}+t) \\ &=2\int_{\Omega}u(t)u_{t}(t)\, dx+2\int _{0}^{t} \bigl(u(\tau),u_{t}(\tau) \bigr) \, d\tau+2\alpha(t_{0}+t) \end{aligned}$$
$$\begin{aligned} H''(t) =&2 \bigl( u_{tt},u(t) \bigr)+2 \bigl\Vert u_{t}(t) \bigr\Vert ^{2}_{2}+2 \bigl(u(t),u_{t}(t) \bigr)+2\alpha \\ =&2 \bigl\Vert u_{t}(t) \bigr\Vert ^{2}_{2}-2M \bigl(\Vert \nabla u\Vert _{2}^{2} \bigr)\Vert \nabla u \Vert _{2}^{2}+2\int_{0}^{t}g(t- \tau)\int_{\Omega}\nabla u(\tau)\nabla u(t)\, dx\, d\tau \\ &{}+2\int _{\Omega}f(u)u\, dx+2\alpha. \end{aligned}$$
$$\begin{aligned}& H(t)H''(t)-(\delta+1)H'(t)^{2} \\& \quad =2H(t) \biggl( \bigl\Vert u_{t}(t) \bigr\Vert ^{2}_{2}-M \bigl(\Vert \nabla u\Vert _{2}^{2} \bigr)\Vert \nabla u\Vert _{2}^{2}+\int _{0}^{t}g(t-\tau)\int_{\Omega}\nabla u(\tau)\nabla u(t)\, dx\, d\tau \\& \qquad {}+\int_{\Omega}f(u)u\, dx+ \alpha \biggr)-4(\delta+1) \biggl(\int_{\Omega}u(t)u_{t}(t) \, dx+\int_{0}^{t} \bigl(u(\tau),u_{t}( \tau) \bigr)\, d\tau+\alpha (t_{0}+t) \biggr)^{2} \\& \quad =2H(t) \biggl( \bigl\Vert u_{t}(t) \bigr\Vert ^{2}_{2}-M \bigl(\Vert \nabla u\Vert _{2}^{2} \bigr)\Vert \nabla u\Vert _{2}^{2}+\int _{0}^{t}g(t-\tau)\int_{\Omega}\nabla u(\tau)\nabla u(t)\, dx\, d\tau \\& \qquad {}+\int_{\Omega}f(u)u\, dx+ \alpha \biggr)+4(\delta+1) \bigl(G(t)- \bigl(H(t)-(T-t)\Vert u_{0} \Vert _{2}^{2} \bigr)\Psi (t) \bigr), \end{aligned}$$
(3.7)
$$\Psi(t)= \bigl\Vert u_{t}(t) \bigr\Vert ^{2}_{2}+ \int_{0}^{t} \bigl\Vert u_{t}(\tau) \bigr\Vert ^{2}_{2}\, d\tau+\alpha $$
$$\begin{aligned} G(t) =&\Psi(t) \biggl( \bigl\Vert u(t) \bigr\Vert _{2}^{2}+ \int_{0}^{t} \bigl\Vert u(\tau) \bigr\Vert ^{2}_{2}\, d\tau+\alpha (t_{0}+t)^{2} \biggr) \\ &{}- \biggl(\int_{\Omega}u(t)u_{t}(t)\, dx+\int _{0}^{t}(u,u_{t})\, d\tau+ \alpha(t_{0}+t) \biggr)^{2}. \end{aligned}$$
$$\begin{aligned}& \biggl(\int_{\Omega}uu_{t}\, dx \biggr)^{2} \leq \bigl\Vert u(t) \bigr\Vert _{2}^{2} \bigl\Vert u_{t}(t) \bigr\Vert ^{2}_{2}, \\& \biggl(\int _{0}^{t}(u,u_{t})\, d\tau \biggr)^{2}\leq \int_{0}^{t} \bigl\Vert u(\tau) \bigr\Vert ^{2}_{2}\, d\tau\int _{0}^{t} \bigl\Vert u_{t}(\tau) \bigr\Vert ^{2}_{2}\, d \tau \end{aligned}$$
$$\begin{aligned}& \int_{\Omega}u(t)u_{t}(t)\, dx\int _{0}^{t} \bigl(u(\tau),u_{t}(\tau) \bigr) \, d\tau \\& \quad \leq \bigl\Vert u(t) \bigr\Vert _{2} \biggl(\int _{0}^{t} \bigl\Vert u_{t}(\tau) \bigr\Vert ^{2}_{2}\, d\tau \biggr)^{\frac {1}{2}} \bigl\Vert u_{t}(t) \bigr\Vert _{2} \biggl(\int_{0}^{t} \bigl\Vert u(\tau) \bigr\Vert ^{2}_{2}\, d\tau \biggr)^{\frac {1}{2}} \\& \quad \leq \frac{1}{2} \bigl\Vert u(t) \bigr\Vert _{2}^{2} \int_{0}^{t} \bigl\Vert u_{t}(\tau) \bigr\Vert ^{2}_{2}\, d\tau+\frac {1}{2} \bigl\Vert u_{t}(t) \bigr\Vert ^{2}_{2}\int _{0}^{t} \bigl\Vert u(\tau) \bigr\Vert ^{2}_{2}\, d\tau \end{aligned}$$
$$\begin{aligned}& \alpha(t+t_{0})\int_{\Omega}u(t)u_{t}(t)\, dx \\& \quad \leq \sqrt{\alpha}\sqrt{\alpha}(t+s_{0}) \bigl\Vert u(t) \bigr\Vert _{2} \bigl\Vert u_{t}(t) \bigr\Vert _{2} \\& \quad \leq \frac{1}{2}\alpha \bigl\Vert u(t) \bigr\Vert _{2}^{2}+ \frac{1}{2}\alpha(t+t_{0})^{2} \bigl\Vert u_{t}(t) \bigr\Vert ^{2}_{2}. \end{aligned}$$
$$ \alpha(t+t_{0})\int_{0}^{t} \bigl( u( \tau), u_{t}(\tau) \bigr)\, d\tau\leq \frac{1}{2}\alpha\int _{0}^{t} \bigl\Vert u(\tau) \bigr\Vert ^{2}_{2}\, d\tau+\frac{1}{2}\alpha(t+t_{0})^{2} \int_{0}^{t}\bigl\Vert u_{t}(\tau)\bigr\Vert ^{2}_{2}\, d\tau. $$
$$ H(t)H''(t)-(\delta+1)H'(t)^{2} \geq H(t)L(t), $$
(3.8)
$$\begin{aligned} L(t) =&-(4\delta+2) \bigl\Vert u_{t}(t) \bigr\Vert ^{2}_{2}-2M \bigl(\Vert \nabla u\Vert _{2}^{2} \bigr)\Vert \nabla u\Vert _{2}^{2} \\ &{}+2 \int_{0}^{t}g(t-\tau)\int_{\Omega}\nabla u(\tau)\nabla u(t)\, dx\, d\tau \\ &{}+2\int_{\Omega}f(u)u\, dx -4(\delta+1)\int _{0}^{t} \bigl\Vert u_{t}(\tau) \bigr\Vert ^{2}_{2}\, d\tau-(4\delta+2)\alpha \\ =&-(4\delta+2) \bigl\Vert u_{t}(t) \bigr\Vert ^{2}_{2}-2 \biggl(M \bigl(\Vert \nabla u\Vert _{2}^{2} \bigr)-\int_{0}^{t} g(\tau)\, d\tau \biggr)\Vert \nabla u \Vert _{2}^{2} \\ &{}+2 \int_{\Omega}f(u)u\, dx -(4 \delta+2)\alpha-4(\delta+1)\int_{0}^{t} \bigl\Vert u_{t}(\tau) \bigr\Vert ^{2}_{2}\, d\tau \\ &{}+2\int _{0}^{t}g(t-\tau)\int_{\Omega}\nabla u(t) \bigl(\nabla u(\tau)-\nabla u(t) \bigr)\, dx\, d\tau. \end{aligned}$$
(3.9)
$$\begin{aligned}& \int_{0}^{t}g(t-\tau)\int _{\Omega}\nabla u(t)\nabla \bigl(u(\tau)-u(t) \bigr)\, dx\, d\tau \\& \quad \geq-(2\delta+1) (g\circ\nabla u) (t)-\frac{1}{(8\delta+4)}\int_{0}^{t}g( \tau)\, d\tau \bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2}. \end{aligned}$$
(3.10)
$$\begin{aligned} L(t) \geq&-(4\delta+2) \bigl\Vert u_{t}(t) \bigr\Vert ^{2}_{2}-(4\delta+2) (g\circ\nabla u) (t) \\ &{}-2 \biggl(M \bigl( \Vert \nabla u\Vert _{2}^{2} \bigr)-\int _{0}^{t} g(\tau)\, d\tau \biggr)\Vert \nabla u \Vert _{2}^{2} \\ &{} +2\int_{\Omega}f(u)u\,dx -4(\delta+1)\int _{0}^{t} \bigl\Vert u_{t}(\tau) \bigr\Vert ^{2}_{2}\, d\tau \\ &{}-\frac{1}{4\delta+2}\int _{0}^{t} g(\tau)\, d\tau \Vert \nabla u\Vert _{2}^{2}-(4\delta+2)\alpha \\ \geq&-(8\delta+4)E(t)+2 \bigl((2\delta+1)\overline{M} \bigl(\Vert \nabla u \Vert _{2}^{2} \bigr)-M \bigl(\Vert \nabla u\Vert _{2}^{2} \bigr)\Vert \nabla u\Vert _{2}^{2} \bigr) \\ &{}-4(\delta+1)\int_{0}^{t} \bigl\Vert u_{t}(\tau) \bigr\Vert ^{2}_{2}\, d\tau -2\int_{\Omega}\bigl[(4\delta+2)F(u)-f(u)u \bigr]\, dx \\ &{}- \biggl(4\delta+\frac{1}{4\delta+2} \biggr) \int_{0}^{t} g(\tau)\, d\tau \Vert \nabla u\Vert _{2}^{2}-(4\delta+2) \alpha \\ \geq&-(8\delta+4)E(0)+2 \bigl((2\delta+1)\overline{M} \bigl(\Vert \nabla u \Vert _{2}^{2} \bigr)-M \bigl(\Vert \nabla u\Vert _{2}^{2} \bigr)\Vert \nabla u\Vert _{2}^{2} \bigr) \\ &{}+4\delta\int_{0}^{t} \bigl\Vert u_{t}(\tau) \bigr\Vert ^{2}_{2}\, d\tau -2\int_{\Omega}\bigl[(4\delta+2)F(u)-f(u)u \bigr]\, dx \\ &{}- \biggl(4\delta+\frac{1}{4\delta+2} \biggr) \int_{0}^{t} g(\tau)\, d\tau \Vert \nabla u\Vert _{2}^{2}-(4\delta+2) \alpha. \end{aligned}$$
$$\begin{aligned} L(t) \geq&-(8\delta+4)E(0)+ \biggl(4\delta m_{0}- \biggl(4\delta+ \frac{1}{4\delta+2} \biggr)\int_{0}^{t} g(\tau)\, d \tau \biggr)\|\nabla u\|_{2}^{2} -(4\delta+2)\alpha \\ \geq& -(8\delta+4)E(0)+ \bigl(4\delta m_{0}- (4\delta+1)l \bigr) \lambda \bigl\Vert u(t) \bigr\Vert _{2}^{2} -(4\delta+2) \alpha \\ \geq&-(8\delta+4)E(0)+ \bigl(4\delta m_{0}- (4\delta+1)l \bigr) \lambda\|u_{0}\|_{2}^{2} -(4\delta+2)\alpha, \end{aligned}$$
$$(8\delta+4)E(0)< \bigl(4\delta m_{0}-(4\delta+2)l \bigr)\lambda \|u_{0}\|_{2}^{2}< \bigl(4\delta m_{0}-(4\delta+1)l \bigr)\lambda\|u_{0} \|_{2}^{2}. $$
$$(4\delta+2)\alpha< \bigl(4\delta m_{0}-(4\delta+2)l \bigr)\lambda \|u_{0}\|_{2}^{2}-(8\delta+4)E(0), $$
$$ L(t)\geq\theta \quad \mbox{for } t\in[0,T]. $$
(3.11)
$$H(t)H''(t)-(\delta+1)H'(t)^{2}>0. $$
$$\alpha t_{0}+\int_{\Omega}u_{0}u_{1} \, dx>0, $$
Setting \(y(t)=H(t)^{-\delta}\), then we have and for all \(t\in[0,T]\), which implies that \(y(t)\) reaches 0 in finite time, say as \(t\rightarrow T^{*}\). Since \(T^{*}\) is independent of the initial choice of T, we may assume that \(T^{*} < T\). This tells us that □
$$y'(t)=-\delta H(t)^{-(\delta+1)}H'(t)< 0 $$
$$y''(t)=-\delta H^{-\delta-2} \bigl(H(t)H''(t)-( \delta+1)H'(t)^{2} \bigr)< 0 $$
$$\lim_{t\rightarrow T^{*}}H(t)=\infty. $$
4 Conclusions
In this paper, we consider the integro-differential equation with initial and Dirichlet boundary conditions which arises in the dynamics of an extensible string with fading memory. Under suitable assumptions on the relax function g and the initial data, we establish a blow-up result with arbitrary positive initial energy. The main tool in proving the blow-up result is the ‘concavity method’ where the basic idea of the method is to construct a positive defined functional \(F(t)\) of the solution by the energy inequality and show that \(F^{-\alpha}(t)\) is a concave function of t.
$$ u_{tt}-M \bigl(\|\nabla u\|^{2}_{2} \bigr)\Delta u+ \int_{0}^{t} g(t-\tau)\Delta u(\tau)\, d\tau+ u_{t}=f(u), \quad (x,t)\in\Omega\times(0,T), $$
Declarations
Acknowledgements
The authors are indebted to the referee for giving some important suggestions which improved the presentation of this paper. This work is supported in part by the scientific research program funded by Shanxi Provincial Education Department No. 14JK1474, the Shanxi Province Postdoctoral Science Foundation, the China Postdoctoral Science Foundation Grant No. 2013M540767, the China NSF Grant No. 11402194 and the doctor scientific research start fund project of Xi’an University of Science and Technology Grant No. 2014QDJ042.
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)
Department of Mathematics, Xi’an University of Science and Technology
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