Abstract
In this article, we investigate a system of two viscoelastic equations with Dirichlet boundary conditions. Under some suitable assumptions on the function \(g_{i}(\cdot )\), \(f_{i}(\cdot ,\cdot )\) (\(i=1,2\)) and the initial data, we obtain general and optimal decay results. Moreover, for certain initial data, we establish a finite time blow-up result. This work generalizes and improves earlier results in the literature. The conditions of the relaxation functions \(g_{1}(t)\) and \(g_{2}(t)\) in our work are weak and seldom appear in previous literature, which is an important breakthrough.
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1 Introduction
In this paper, we investigate the following initial-boundary problem:
where Ω is a bounded domain of \(R^{n}\) (\(n\geq 1\)) with a smooth boundary ∂Ω, u and v represent the transverse displacements of wave. The functions \(g_{1}\) and \(g_{2}\) denote the kernel of the memory term, and the nonlinear functions \(f_{1}\) and \(f_{2}\) will be specified later. Problem (1.1) describes the propagation of some material possessing a capacity of storage and dissipation of mechanical energy. Models of this type arise in the theory of viscoelasticity and physics.
Firstly, we present some results related to viscoelastic equations. The single viscoelastic equation of the form
which has been extensively studied by many authors, and related results concerning existence, decay, and blow-up have been recently established (see [6, 11]). Here, we understand \(\triangle u_{tt}\), \(\vert u _{t} \vert ^{m-2}u_{t}\), \(\int _{0}^{t}g(t-\tau )\bigtriangleup u(\tau )d \tau \), and \(\vert u \vert ^{p-2}u\) to be the dispersion term, weak damping term, viscoelasticity dissipative term, and source term, respectively. This type of problem usually appears as a model in nonlinear viscoelasticity.
As \(a=b=0\) in the presence of the strong damping term, Cavalcanti et al. [1] dealt with the equation
where Ω is a bounded domain with smooth boundary ∂Ω in \(R^{n}\) (\(n\geq 1\)) and \(\rho >0\). They established a global existence result when the constant \(\gamma \geq 0\) and an exponential decay result for the case \(\gamma >0\). Later, this result was improved by Messaoudi and Tatar [8] to a situation where a source term is presented. By using perturbation techniques, they established a global existence and an exponential decay result.
When \(a=b=1\) in the absence of the dispersion term, the following problem
in a bounded domain and \(m>2\), \(p>2\) was studied by Song [10]. The author proved the nonexistence of global solutions of problem (1.4) with bounded positive initial energy.
The case of \(\rho =0\) in the absence of the viscoelasticity term problem (1.2) has been discussed by many authors. For example, Chen and Liu [2] studied the following equation:
subject to the same boundary and initial conditions as problem (1.4). Under some suitable conditions on the initial data, they proved the global existence of solutions in both cases which are polynomial and exponential decay in the energy space respectively.
In the case of \(a=b=1\), \(m=2\), in the presence of the strong damping term and without the dispersion term, the following viscoelastic equation
was studied by Li and He [4]. By using some properties of the convex functions, they obtained a general decay rate result. Moreover, they established a finite time blow-up result for solutions with negative initial energy and positive initial energy.
For a coupled system, the following system of viscoelastic equations
was considered by Mustafa [9]. Under some suitable conditions, they proved the well-posedness and obtained a generalized stability result.
In the work [5], Liu considered the following problem:
where \(\gamma _{1},\gamma _{2}>0\) and \(0<\rho \leq \frac{2}{n-2}\) if \(n\geq 3\) or \(\rho >0\) if \(n=1,2\). Under the following assumptions on the relaxation functions:
where \(\zeta _{1}\) and \(\zeta _{2}\) are positive constants. By exploiting the perturbed energy method, a uniform decay of the energy result was established.
Recently, Houari et al. [3] investigated the following system:
where Ω is an open bounded domain of \(R^{n}\) with a smooth boundary ∂Ω, m, \(r\geq 1\). Under some suitable conditions, they obtained a general decay, which depends on the relaxation functions.
As for a single wave equation, in the absence of the source term, the damping term assures global existence. On the other hand, without the damping term, the source term may cause finite time blow-up of solution. Hence, it is valuable to study the viscoelastic equation with damping and source terms.
Our aim in this work is to establish the global existence, decay, and blow-up result of solutions to problem (1.1). By adopting and modifying the methods used in [9], we establish the general and optimal decay and blow-up results, while we should overcome the additional difficulty caused by the changes of the conditions of the relaxation functions \(g_{1}(t)\) and \(g_{2}(t)\). In [9], the relaxation functions \(g_{i}(t)\) (\(i=1,2\)) satisfy \(g_{i}'(t)\leq - \zeta _{i}g_{i}(t)\) for all \(t\geq 0\), where \(\zeta _{i}(t)\) are positive nonincreasing functions. In this paper, the conditions have been replaced by \(g_{i}'(t)\leq -\zeta _{i}(t)g_{i}^{\gamma }(t)\), \(\gamma \in [1, 3/2)\). As far as we know, the conditions of the relaxation functions \(g_{i}(t)\) in our work seldom appear in previous literature, which is an important breakthrough. Our main novel contribution is an extension and improvement of the previous result from [9].
This paper is organized as follows. In Sect. 2, we give material needed for our work. In Sect. 3, we prove the global existence. In Sect. 4, we present some technical lemmas needed in the proof of our result. Section 5 is devoted to the general decay result. In Sect. 6, we carry out the proof of finite time blow-up result.
2 Preliminaries
In this section, we present some material needed for our work. First, we make the following assumptions.
-
(G1)
\(g_{i}: \mathbb{R_{+}}\rightarrow \mathbb{R_{+}}\) (for \(i=1,2\)) are \(C^{1}\) nonincreasing functions satisfying
$$ 1- \int _{0}^{\infty }g_{1}(\tau )\,d\tau =l>0,\quad\quad 1- \int _{0}^{\infty }g_{2}(\tau )\,d\tau =k>0, \quad g_{i}(0)>0. $$(2.1) -
(G2)
There exist two nonincreasing differentiable functions \(\zeta _{1}, \zeta _{2}:\mathbb{R_{+}}\rightarrow \mathbb{R_{+}}\), with \(\zeta _{1}(0), \zeta _{2}(0)>0\) and satisfying
$$ g_{i}'(t)\leq -\zeta _{i}(t)g_{i}^{\gamma }(t), \quad t\geq 0, 1\leq \gamma < \frac{3}{2}, \text{for }i=1,2. $$(2.2) -
(G3)
For the functions \(f_{1}\) and \(f_{2}\), we note that
$$\begin{aligned}& f_{1}(u,v)= \vert u+v \vert ^{2(p+1)}(u+v)+ \vert u \vert ^{p}u \vert v \vert ^{p+2}, \end{aligned}$$(2.3)$$\begin{aligned}& f_{2}(u,v)= \vert u+v \vert ^{2(p+1)}(u+v)+ \vert u \vert ^{p+2}v \vert v \vert ^{p}, \end{aligned}$$(2.4)where p satisfies
$$ p>-1 \quad \text{if } n=1,2 \quad \text{and} \quad -1< p\leq \frac{3-n}{n-2} \quad \text{if } n\geq 3. $$(2.5)
It is easy to verify that
where
Remark 2.1
There are many functions \(g_{i}(t)\) and \(\zeta _{i}(t)\) (for \(i=1,2\)) satisfying (G1) and (G2). An example of such functions is
Definition 2.2
A pair of functions \((u,v)\) defined on \([0,T]\) is called a weak solution of system (1.1) if \(u,v\in C_{w}([0,T],H_{0}^{1}(\varOmega ))\), \(u _{t},v_{t}\in C_{w}([0,T],L^{2}(\varOmega ))\), \((u(x,0),v(x,0))=(u_{0},v _{0})\in H_{0}^{1}(\varOmega )\times H_{0}^{1}(\varOmega )\), \((u_{t}(x,0),v _{t}(x,0))=(u_{1},v_{1})\in H_{0}^{1}(\varOmega )\times H_{0}^{1}(\varOmega )\) and \((u,v)\) satisfies
for a.e. \(t\in [0,T]\) and all test functions \(\phi ,\psi \in H_{0} ^{1}(\varOmega )\). Here, \(C_{w}([0,T],X)\) denotes the space of weakly continuous functions from \([0,T]\) into Banach space X.
Proposition 2.3
Let \((u_{0},v_{0})\in H_{0}^{1}(\varOmega )\times H_{0}^{1}(\varOmega )\) and \((u_{1},v_{1})\in L^{2}(\varOmega )\times L^{2}(\varOmega )\). Assume that (G1)–(G3) hold. Then there exists a unique local weak solution \((u,v)\) of system (1.1) defined in \([0,T_{m}]\) for some \(T_{m}>0\) small enough.
3 Global existence
In this section, we prove the global existence of the solution to system (1.1).
First, we define
and the energy function associated with system (1.1)
where
Lemma 3.1
Let (G1)–(G3) hold and \((u,v)\) be the solution of (1.1), then we have
Proof
Multiplying the first equation in system (1.1) by \(u_{t}\) and integrating over Ω gives
Then
For the last term on the left-hand side of (3.8), we get
By combining (3.4), (3.8), and (3.9), we deduce
Similarly, multiplying the second equation in (1.1) by \(v_{t}\) and integrating over Ω yields
Finally, by adding (3.10) to (3.11), (3.6) is established. □
The following lemma is important to prove the global existence of solution.
Lemma 3.2
([3])
Assume that (2.5) holds. Then there exists \(\eta >0\) such that, for any \((u,v)\in H_{0}^{1}(\varOmega )\times H_{0}^{1}(\varOmega )\), we obtain
Proof
Exploiting Minkowski’s inequality, we have
From Hölder’s inequality and Young’s inequality, we derive
Then, combining (3.13), (3.14) and the embedding \(H_{0}^{1}( \varOmega )\hookrightarrow L^{2(p+2)}(\varOmega )\) leads to (3.12). □
Lemma 3.3
Let \((u_{0},v_{0})\in H_{0}^{1}(\varOmega )\times H_{0}^{1}(\varOmega )\) and \((u_{1},v_{1})\in L^{2}(\varOmega )\times L^{2}(\varOmega )\). Assume that (G1)–(G3) hold. If
then \(I(t)>0\) for \(t\in [0,T_{m}]\).
Proof
Since \(I(0)>0\), then by the continuity of \(I(t)\), there exists \(T^{*}< T_{m}\) such that \(I(t)>0\), \(\forall t\in [0,T^{*}]\). By using (3.1) and (3.2), we have
From (2.1), (3.1), (3.2), and (3.6), we infer that
It follows from (2.7), (3.12), and (3.15) that
Combining (3.15) and (3.18), we deduce that
Therefore
By repeating this procedure, \(T^{*}\) is extended to \(T_{m}\). □
Lemma 3.4
Assume that (G1)–(G3) hold. If \((u_{0},v_{0})\in H_{0} ^{1}(\varOmega )\times H_{0}^{1}(\varOmega )\) and \((u_{1},v_{1})\in L^{2}( \varOmega )\times L^{2}(\varOmega )\) and satisfy (3.15), then the solution of system (1.1) is bounded and global in time.
Proof
From Lemma 3.3, (3.6), and (3.17), we see that
therefore
which implies that the solution of system (1.1) is global and bounded. □
4 Technical lemmas
In this section, we present some lemmas needed for the proof of our result.
Lemma 4.1
([3])
There exist two constants \(a_{0}>0\) and \(a_{1}>0\) such that
Proof
The right-hand side of (4.1) is trivial. If \(u=v=0\), for the left-hand side of (4.1), the result is also trivial. If, without loss of generality, \(v\neq 0\), then either \(\vert u \vert \leq \vert v \vert \) or \(\vert u \vert > \vert v \vert \).
If \(\vert u \vert \leq \vert v \vert \), we have
Now we consider the continuous function
then we obtain that \(\min j(\omega )\geq 0\). If \(\min j(\omega )=0\), then there exists \(\omega _{0}\in [-1,1]\) such that
This infers that \(\vert 1+\omega _{0} \vert = \vert \omega _{0} \vert =0\), which is impossible. Hence \(\min j(\omega )=2a_{0}>0\). Thus
It follows from (4.5) that
and then
If \(\vert u \vert > \vert v \vert \), we deduce that
Hence, this gives the desired result. □
Lemma 4.2
([3])
There exist two positive constants \(\lambda _{1}\) and \(\lambda _{2}\) such that
Proof
From (2.3), we easily get
By (4.10) and Young’s inequality, with \(q=\frac{2p+3}{p+1}\) and \(q'=\frac{2p+3}{p+2}\), we have
therefore
Then, by using Poincare’s inequality and (2.5), we observe that
In the same way, we have
□
Lemma 4.3
(Jensen inequality)
Suppose that G is a concave function on \([a,b]\), \(f:\varOmega \rightarrow [a,b]\) and h are integrable functions on Ω, with \(h(x) \geq 0\), and \(\int _{\varOmega }h(x)\,dx=r>0\), then
For the special case \(G(y)=y^{\frac{1}{\gamma }}\), \(y\geq 0\), \(\gamma >1\), we obtain
Lemma 4.4
Assume that \(g_{i}\) satisfies (G1) and (G2) for \(i=1,2\), then
Proof
From (G1) and (G2), we see that
Integrating (4.18) over \((0,+\infty )\) and using the fact that \(0\leq \theta <2-\gamma \), we get
□
Lemma 4.5
([7])
If (G1)–(G3) hold, \(u\in L^{\infty }(0,T,H_{0}^{1}(\varOmega ))\), for \(0<\theta <1\), we obtain
By taking \(\theta =\frac{1}{2}\), we have
Proof
For \(q>1\), we derive
Applying Hölder’s inequality, we deduce
By taking \(q=\frac{\gamma -1+\theta }{\gamma -1}\), we arrive at
Then, by taking \(\theta =\frac{1}{2}\) in (4.24), (4.21) is established. □
Similarly,
and
Lemma 4.6
([7])
Suppose that \(g_{1}\) satisfies (G1) and (G2), then we have
Proof
Multiplying both sides of (4.21) by \(\zeta _{1}(t)\) and by using (G2), (3.6), and (4.17) gives
□
Likewise,
5 The decay result
In this section, we establish three related lemmas before proving our result.
Lemma 5.1
If (G1)–(G3) and (3.15) hold, the functional \(\phi (t)\) defined by
satisfies, along solutions of (1.1),
Proof
A differentiation of \(\phi (t)\) with respect to t, it follows from system (1.1) that
For the first term on the right-hand side of (5.2), by using Young’s inequality and the fact that \(\int _{0}^{t}g_{1}(s)\,ds\leq \int _{0}^{\infty }g_{1}(s)\,ds =1-l\), for \(\eta =\frac{l}{1-l}>0\), we get
Similar calculations also yield, for \(\eta _{1}=\frac{k}{1-k}>0\),
Applying Young’s inequality and Poincare’s inequality, for some \(\beta _{1}>0\), we obtain
Likewise, for some \(\beta _{2}>0\), we have
Inserting (5.3)–(5.6) into (5.2) yields
Now, we pick \(\beta _{1}, \beta _{2}>0\) small enough such that
Finally, a combination of (5.7) and (5.8) gives (5.1). □
Lemma 5.2
Suppose that (G1)–(G3) and (3.15) hold. The functional \(\psi _{1}(t)\) defined by
satisfies, along solutions of (1.1),
where \(\alpha _{2}=\lambda _{1} (\frac{2(p+2)}{p+1}E(0) )^{2p+2}\), \(\lambda _{1}\) is the constant in Lemma 4.2.
Proof
Taking the derivative of \(\psi _{1}(t)\) with respect to t and using system (1.1) gives
For the first term on the right-hand side of (5.10), by exploiting (G1), Young’s inequality, and Cauchy–Schwarz inequality, for any \(\delta >0\), we get
As for the second term in (5.10), recall that \((a+b)^{2}\leq 2(a ^{2}+b^{2})\), for \(\eta _{2}=1\), we obtain
The third term can be handled by
For the forth term, it follows from (4.9) that
where \(\alpha _{2}=\lambda _{1} (\frac{2(p+2)}{p+1}E(0) )^{2p+2}\).
The fifth term on the right-hand side of (5.10) can be estimated as
Taking into account estimates (5.11)–(5.13), estimate (5.9) is established. □
Similar computations also yield the following.
Lemma 5.3
Suppose that (G1)–(G3) and (3.15) hold. The functional \(\psi _{2}(t)\) defined by
satisfies, along solutions of system (1.1),
where \(\alpha _{3}=\lambda _{2} (\frac{2(p+2)}{p+1}E(0) )^{2p+2}\), \(\lambda _{2}\) is the constant in Lemma 4.2.
Now, we define the functional
where \(\epsilon _{1}\) and \(\epsilon _{2}\) are positive constants, \(\phi (t)\) is given in Lemma 5.1 and \(\psi (t):=\psi _{1}(t)+\psi _{2}(t)\).
Lemma 5.4
([5])
Let \((u,v)\) be the solution of system (1.1) and assume that (3.15) holds. Then there exist constant \(\epsilon >0\) small enough and \(M>0\) large enough such that the following relation
holds for two positive constants \(\beta _{1}\) and \(\beta _{2}\).
Theorem 5.5
Assume (G1)–(G3) and (3.15) hold. Let \((u_{0},v_{0}) \in H_{0}^{1}(\varOmega )\times H_{0}^{1}(\varOmega )\) and \((u_{1},v_{1}) \in L^{2}(\varOmega )\times L^{2}(\varOmega )\). Then, for each \(t_{0}>0\), there exist positive constants K, k, \(k_{1}\), \(k_{2}\) such that the solution of system (1.1) satisfies, for all \(t\geq t_{0}\),
Furthermore, if
then
where \(\zeta (t)=\min \{\zeta _{1}(t),\zeta _{2}(t) \}\).
Proof
From (G1), we know \(g_{1}\) and \(g_{2}\) are positive, then for any \(t\geq t_{0}>0\), we have
Taking derivative of (5.17) with respect to t and using (3.6), (5.1), and (5.16) yields
At this point, we pick \(\delta >0\) small enough such that
and
As long as δ is fixed, the choice of any two positive constants \(\epsilon _{1}\) and \(\epsilon _{2}\) satisfying
and
will make
Hence, there exist two positive constants m and C such that
Case 1. \(\gamma =1\):
Let \(\zeta (t)=\min \{\zeta _{1}(t),\zeta _{2}(t) \}\), since \(\zeta _{1}(t)\) and \(\zeta _{2}(t)\) are nonincreasing differentiable functions, then we get that \(\zeta (t)\) is nonincreasing.
When \(\gamma =1\), from (2.2), we easily get
Multiplying both sides of (5.24) by \(\zeta (t)\) and applying (3.6), (4.27), (4.29), and (5.25) yields
Setting \(F(t)=\zeta (t)W(t)+CE(t)\), then clearly \(F\sim E\). Recalling that \(W\sim E\geq 0\) by (3.21) and (5.18), \(\zeta '(t) \leq 0\) by (G2), we get that \(\zeta '(t)W(t)\leq 0\), then together with (5.26), we have, for some \(k>0\),
Integrating (5.27) over \([t_{0},t]\) gives
Therefore, by using the fact that \(F(t)\sim E(t)\), we derive
Case 2. \(1<\gamma <\frac{3}{2}\):
Multiplying both sides of (5.24) by \(\zeta (t)\), using (4.27) and (4.29) leads to
Multiplying (5.30) by \(\zeta ^{\alpha }(t)E^{\alpha }(t)\), with \(\alpha =2\gamma -2\), we get
Exploiting Young’s inequality with \(q=1+\alpha \) and \(q'=\frac{1+ \alpha }{\alpha }\) gives
We pick \(\epsilon <\frac{m}{C}\) and recall that \(\zeta '(t)\leq 0\), \(\zeta (t)>0\) by (G2), \(E'(t)\leq 0\) by (3.6), and \(W\thicksim E\geq 0\) by (3.21) and (5.18), then together with (5.32) we have, for some \(c_{1}>0\),
Next, we take \(F(t)=\zeta ^{\alpha +1}WE^{\alpha }+CE\), which is clearly equivalent to \(E(t)\), then there exists \(a_{0}>0\) such that
Integrating (5.34) over \((t_{0},t)\) and recalling that \(F(t)\thicksim E(t)\) and \(\alpha =2\gamma -2\), we obtain
From (5.21) and (5.35), we infer that
Setting \(\lambda _{1}(t)=\int _{0}^{t} \Vert \nabla u(t)-\nabla u(t-\tau ) \Vert _{2} ^{2}\,d\tau \), by using (3.3), we deduce
Similarly, let \(\lambda _{2}(t)=\int _{0}^{t} \Vert \nabla v(t)-\nabla v(t- \tau ) \Vert _{2}^{2}\,d\tau \), we have \(\lambda _{2}(t)<+\infty \).
From (5.24) and recalling that \(\zeta (t)=\min \{\zeta _{1}(t), \zeta _{2}(t) \}\), \(\zeta _{1}(t)\) and \(\zeta _{2}(t)\) are nonincreasing differentiable functions, we arrive at
Exploiting Jensen’s inequality, for the second term on the right-hand side of (5.38), with \(G(y)=y^{\frac{1}{\gamma }}\), \(y>0\), \(f(\tau )= \zeta ^{\gamma }(\tau )g^{\gamma }(\tau )\) and \(h(\tau )= \Vert \nabla u(t)- \nabla u(t-\tau ) \Vert _{2}^{2}\), where we assume that \(\lambda _{1}(t), \lambda _{2}(t)>0\), otherwise we get \(\Vert \nabla u(t)-\nabla u(t-\tau ) \Vert = \Vert \nabla v(t)-\nabla v(t-\tau ) \Vert =0\), then by using (5.24) we deduce
Since \(\zeta _{1}(t)\) and \(\zeta _{2}(t)\) are nonincreasing, then for some \(C_{1}>0\), estimate (5.38) becomes
Multiplying both sides of (5.39) by \(\zeta ^{\alpha }(t)E^{\alpha }(t)\), with \(\alpha =\gamma -1\), we deduce
Applying Young’s inequality, with \(q=1+\alpha \), and \(q'=\frac{1+ \alpha }{\alpha }\) leads to
Then, by taking \(\sigma <\frac{m}{C_{1}}\) and recalling that \(\zeta '(t)\leq 0\), \(\zeta (t)>0\) by (G2), \(E'(t)\leq 0\) by (3.6), and \(W(t)\thicksim E(t)\geq 0\) by (3.21) and (5.18), together with (5.41), we get, for some \(C_{2}>0\),
which implies
Let \(L=\zeta ^{\alpha +1}E^{\alpha }W+CE\), then clearly \(L\thicksim E\), we obtain, for some \(C_{3}>0\),
Integrating (5.44) over \((t_{0},t)\) and recalling that \(L\thicksim E\) and \(\alpha =\gamma -1\) yields
This completes the proof. □
6 The blow-up result
In this section, we carry out the proof of the finite time blow-up result.
Theorem 6.1
If (G1) and (G3) hold and the initial energy \(E(0)<0\). Assume that \(g_{i}\) satisfies
Then the solution of system (1.1) blows up in finite time.
Proof
First, we define
where \(E(t)\) is defined in (3.3). By (G1) and (3.6), we find that
Noting the assumption \(E(0)<0\), from (4.1), (6.2), and (6.3), we get
From (4.1), we see that
Let
where ϵ is a positive constant to be chosen later and
Taking the derivative of \(L(t)\) and using system (1.1) gives
It follows from Young’s inequality that
Similarly,
A combination of (6.8)–(6.10) leads to
Let \(0< r\leq \min \{l, k \}\), then by the expression of \(E(t)\) and \(H(t)\), we obtain
Inserting (6.12) into (6.11), we arrive at
By using (G1), (6.1), and \(r=\min \{l, k \}\), we infer that
For the last two terms of (6.13), by using Hölder’s inequality and (6.3)–(6.5) yields
where \(K_{1}=c_{0}^{\frac{-1}{2(p+2)}} \vert \varOmega \vert ^{\frac{p+1}{2(p+2)}}\).
By (6.7), we know \(\frac{1}{2(p+2)}-\frac{1}{2}+\alpha <0\), it follows from Young’s inequality and (6.3) that
Likewise,
We pick
then it follows from (6.13)–(6.18) that
where
Now, we choose \(0<\epsilon <1\) small enough such that
A combination of (6.14), (6.19), and (6.21) yields
where C is a positive constant. It is clear that \(L(t)\) is increasing on \([0,T)\) and
In the case of \(\int _{\varOmega }u_{0}u_{1}\,dx+\epsilon \int _{\varOmega }v _{0}v_{1}\,dx\geq 0\), no further restriction on ϵ is needed. For the case \(\int _{\varOmega }u_{0}u_{1}\,dx+\epsilon \int _{\varOmega }v_{0}v _{1}\,dx<0\), we assume that
Therefore, in either case, we have
Now we prove that \(L(t)\) satisfies the following inequality:
where C is a positive constant, \(1<\frac{1}{1-\alpha }<2\) assumed by (6.7). At this point, we distinguish two cases.
Case 1. \(\int _{\varOmega }uu_{t}\,dx+\int _{\varOmega }vv_{t}\,dx\leq 0\) for some \(t\in [0,T)\). For such t, we obtain
which together with (6.22), then (6.26) follows for all such t.
Case 2. \(\int _{\varOmega }uu_{t}\,dx+\int _{\varOmega }vv_{t}\,dx\geq 0\) for all \(t\in [0,T)\). Since \(0<\alpha <\frac{1}{2}\), \(1<\frac{1}{1-\alpha }<2\), and \(0<\epsilon <1\), then we deduce
Exploiting Hölder’s inequality and Young’s inequality, we see that
From (6.7), we get that \(\frac{1}{(1-2\alpha )(p+1)}-1<0\). Then, by (6.3) and (6.5), we have
Similarly,
By combining (6.28)–(6.31), we deduce that
It follows from (6.22) and (6.32) that
which shows (6.26) follows for Case 2.
Integrating (6.26) over \((0,t)\) yields
This shows that \(G(t)\) blows up in finite time
Therefore, the solution of system (1.1) blows up in finite time. □
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He, L. On decay and blow-up of solutions for a system of viscoelastic equations with weak damping and source terms. J Inequal Appl 2019, 200 (2019). https://doi.org/10.1186/s13660-019-2155-y
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DOI: https://doi.org/10.1186/s13660-019-2155-y