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On the decay and blow-up of solution for a system of nonlinear viscoelastic plate equations with dissipative terms
Boundary Value Problems volume 2016, Article number: 115 (2016)
Abstract
In this paper, we consider the initial-boundary value problem of nonlinear viscoelastic plate equations with dissipative terms. We prove that, for certain initial data in the stable set, the decay rate estimate of the energy function is exponential or polynomial depending on the exponents of the damping terms in both equations by using Nakao’s method. Conversely, for certain initial data in the unstable set, we use the perturbed energy method to show that the solution blows up in finite time when the initial energy is not larger than some positive number. This improves earlier results in the literature.
1 Introduction
In this paper, we consider the following initial-boundary value problem of the nonlinear viscoelastic plate equations with dissipative terms:
where Ω is a bounded domain in \(R^{n}\) (\(n=1,2,3\)) with smooth boundary ∂Ω, γ and δ are positive constants, \(g_{i}: R^{+}\rightarrow R^{+}\), \(f_{i}: R^{2}\rightarrow R\), \(i=1,2\), are given functions to be specified later.
The motivation of our work is due to the initial boundary problem of the plate equation
which has been discussed by Di and Shang [1] by considering the existence of global solutions and the asymptotic behavior of global solutions with \(m\geq p\). Here, we understand \(\Delta u_{t}\), \(-\Delta u_{tt}\), \(a|u_{t}|^{m-2}u_{t}\), and \(b|u|^{p-2}u\) to be the strong dissipation term, the dispersive term, the nonlinear damping term, and the source term, respectively.
In the absence of the dispersive term and the nonlinear damping term, model (1.2) reduces to the following wave equation (\(n\geq1\))
In 2000, Shang [2] studied the well-posedness, asymptotic behavior, and the finite time blow-up of the solutions under some suitable conditions on f and for \(n=1,2,3\). In 2004, Zhang and Hu [3] showed the existence and the stability of global weak solutions. In 2007, Xie and Zhong [4] obtained the existence of global attractors in \(H_{0}^{1}(\Omega)\times H_{0}^{1}(\Omega)\), where the nonlinear term f satisfies a critical exponential growth assumption. In 2008, Xu et al. [5] used the multiplier method to investigate the asymptotic behavior of solutions for (1.3). Kafini and Messaoudi [6] considered a nonlinear wave equation and obtained a finite-time blow-up result with arbitrary positive initial energy. For more related results, the reader is referred to [7–10].
In the absence of the dispersive term and \(m=0\), model (1.2) reduces to the wave equation
Xu and Yang [11] established a blow-up result for certain solutions of (1.4) with arbitrary positive initial energy, where \(1< p<\infty\) if \(n=1,2\) and \(1< p\leq\frac{n+2}{n-2}\) if \(n\geq3\).
Messaoudi and Mukiawa [12] studied the fourth-order viscoelastic plate equation
in the bounded domain \(\Omega=(0, \pi)\times(-l, l)\subset R^{2}\) with nontraditional boundary conditions. The authors established the well-posedness of the solution and a decay result.
Another model related to (1.1) is
where Ω is a bounded domain in \(R^{n}\) (\(n=1,2,3\)) with smooth boundary ∂Ω. For problem (1.5) with \(h_{1}(u_{t})=-\Delta u_{t}\) and \(h_{2}(v_{t})=-\Delta v_{t}\), Liang and Gao [13] obtained that the decay estimate of the energy function is exponential with certain initial data in the stable set. On the contrary, a solution with positive initial energy blows up in finite time when the initial data is inside the unstable set. For \(h_{1}(u_{t})=|u_{t}|^{m-1}u_{t}\) and \(h_{2}(v_{t})=|v_{t}|^{r-1}v_{t}\), Han and Wang [14] showed several results concerned with local existence, global existence, and finite-time blow-up with negative initial energy. The latter blow-up result has been improved by Messaoudi, Said-Houari, and Guesmia [15, 16] by studying a larger class of initial data for which the initial energy can take positive values and obtained that the rate of decay of the total energy depends on those of the relaxation functions. Wu [17] considered the following problem for \((x,t)\in\Omega\times(0,T)\):
where Ω is a bounded domain in \(R^{n}\) (\(n=1,2,3\)) with smooth boundary ∂Ω. He obtained that the decay estimate of the energy function is exponential or polynomial depending on the exponents of the damping terms in both equations, and the blow-up of solution with nonnegative initial energy was established.
Motivated by previous works, it is interesting to study the global existence, uniform decay, and finite time blow-up of solution to problem (1.1). Firstly, we establish that the solution is global in time under certain initial data in the stable set. After that, we show the decay estimate of solutions by Nakao’s method [18]. Precisely, we establish that the decay estimate of energy function is exponential or polynomial depending on the parameters p and q. Secondly, we study the finite time blow-up of problem (1.1) with \(\gamma=\delta=1\). By adopting and modifying the methods used in [15] we prove the blow-up of solutions when the energy is negative or nonnegative and less than the critical value \(E_{1}\) (given in (4.3)). In this way, our results allow a wider region for the blow-up results.
The paper is organized as follow. In Section 2, we present preliminaries and some lemmas. In Section 3, the global existence and decay property are derived. Finally, the blow-up results of (1.1) with \(\gamma=\delta=1\) are obtained in the case of initial energy being nonnegative.
2 Preliminaries
In this section, we give some lemmas and assumptions. We use the standard Lebesgue space \(L^{p}(\Omega)\) and Sobolev space \(H_{0}^{1}(\Omega)\) with their usual products and norms. We use the embedding \(H_{0}^{1}(\Omega)\hookrightarrow L^{p}(\Omega)\) for \(2\leq p\leq\frac{2n}{n-2}\) if \(n\geq3\) or \(2\leq p\) if \(n=1,2\). In this case, the embedding constant is denoted by \(c_{*}\), that is,
Next, we give the assumptions for problem (1.1).
-
(A1)
The relaxation functions \(g_{1}(s)\) and \(g_{2}(s)\) are of class \(C^{1}\), nonnegative and nonincreasing for \(s\geq0\), and satisfy
$$\gamma- \int_{0}^{\infty}g_{1}(s)\,ds=l>0,\qquad \delta- \int_{0}^{\infty}g_{2}(s)\,ds=k>0. $$Concerning the functions \(f_{1}(u,v)\) and \(f_{2}(u,v)\), we take (see [15])
$$ f_{1}(u,v)=(m+1) \bigl(a|u+v|^{m-1}(u+v)+b|u|^{\frac{m-3}{2}}|v|^{\frac {m+1}{2}}u \bigr) $$(2.2)and
$$ f_{2}(u,v)=(m+1) \bigl(a|u+v|^{m-1}(u+v)+b|v|^{\frac{m-3}{2}}|u|^{\frac {m+1}{2}}v \bigr) $$(2.3)with constants \(a,b>0\). We can easily verify that
$$uf_{1}(u,v)+vf_{2}(u,v)=(m+1)F(u,v),\quad (u,v)\in R^{2}, $$where
$$F(u,v)=a| u+v|^{m+1}+2b| uv|^{\frac{m+1}{2}}. $$ -
(A2)
For the nonlinearity, we suppose that
$$ \left \{ \textstyle\begin{array}{l@{\quad}l} m>1,& n=1,2, \\ 1< m\leq3, & n=3, \end{array}\displaystyle \right . $$(2.4)and
$$ \left \{ \textstyle\begin{array}{l@{\quad}l} p,q\geq1,& n=1,2, \\ 1\leq p,q\leq5,& n=3. \end{array}\displaystyle \right . $$(2.5)
As in [15], we still have the following results.
Lemma 2.1
(Sobolev-Poincaré inequality)
Let \(2\leq k<+\infty\) and \(n\leq3\). Then there is a constant \(\tilde{c}=c(\Omega ,k)\) such that
Lemma 2.2
There exist two positive constants \(c_{0}\) and \(c_{1}\) such that
Lemma 2.3
Suppose that (2.4) holds. Then there exists \(\eta>0\) such that, for any \((u,v)\in H_{0}^{1}(\Omega)\times H_{0}^{1}(\Omega)\), we have
We also need the following technical lemma.
Lemma 2.4
([15])
For any \(g\in C^{1}\) and \(\phi\in H^{1}(0,T)\), we have
where
Now, we are in a position to state the local existence result to problem (1.1), which can be established by using arguments similar to those in [14]. We omit the proof.
Theorem 2.5
Let \(u_{0},v_{0}\in H_{0}^{2}(\Omega)\) and \(u_{1},v_{1}\in H_{0}^{1}(\Omega)\). Assume that (A1) and (A2) are satisfied. Then there exists a couple solution \((u,v)\) of problem (1.1) such that, for some \(T>0\),
We conclude this section by stating Nakao’s lemma, which will be used in establishing the decay rate of solutions to problem (1.1).
Lemma 2.6
([18])
Let \(\phi(t)\) be a nonincreasing and nonnegative function on \([0,T]\), \(T>1\), such that
where \(\omega_{0}>1\) and \(r\geq0\). Then we have, for all \(t\in[0,T]\),
-
(i)
if \(r=0\), then
$$\phi(t)\leq\phi(0)e^{-\omega_{1}[t-1]^{+}}, $$ -
(ii)
if \(r>0\), then
$$\phi(t)\leq \bigl(\phi^{-r}(0)+\omega_{0}^{-1}r[t-1]^{+} \bigr)^{-\frac{1}{r}}, $$
where \(\omega_{1}:=\ln(\frac{\omega_{0}}{\omega_{0}-1})\) and \([t-1]^{+}:=\max\{t-1,0\}\).
Remark 2.7
For simplicity, we take \(a=b=1\) in (2.2) and (2.3) throughout this paper.
3 Global existence and energy decay
In this section, we focus our attention on the global existence and decay rate of the solution to problem (1.1). We first define
and define the energy function as
Lemma 3.1
Suppose that (A1) and (2.4) hold. Let \((u,v)\) be the solution of problem (1.1). Then \(E(t)\) is a nonincreasing function, that is, for \(t\geq0\),
Proof
Multiplying (1.1)1 by \(u_{t}\) and (1.2)2 by \(v_{t}\), integrating over Ω, summing up, and then using integration by parts, we obtain
Applying Lemma 2.4 to the third and fourth terms on the right-hand side of this equality, we get (3.4). □
Lemma 3.2
Suppose that (A1) and (2.4) hold. Let \((u,v)\) be the solution of problem (1.1). Assume further that \(I(0)>0\) and
Then
Proof
Since \(I(0)>0\), by continuity there exists a maximal time \(t_{\max}>0\) (possibly \(t_{\max}=T\)) such that
which implies that, for \(t\in[0,t_{\max}]\),
Since \(E(t)\) is nonincreasing by (3.4), using (3.7) and (3.3), we get, for \(t\in[0,t_{\max}]\),
Using Lemma 2.2, (2.6), (3.8), and (3.5), we obtain, for \(t\in [0,t_{\max}]\),
Thus,
By repeating these steps and using the fact that
this implies that we can take \(t_{\max}=T\). □
Lemma 3.3
Let the assumptions of Lemma 3.2 hold. Then there exists \(\eta_{1}>1\) such that
where \(\eta_{1}= {\frac{1}{1-\alpha_{1}}}\).
Proof
From (3.9) we have
Letting \(\eta_{1}= {\frac{1}{1-\alpha_{1}}}\) and using (3.1), we obtain (3.10). □
Theorem 3.4
Suppose that (A1) and (A2) hold. Let \(u_{0},v_{0}\in H_{0}^{2}(\Omega)\) and \(u_{1},v_{1}\in H_{0}^{1}(\Omega )\) satisfy \(I(0)>0\) and (3.5). Then the solution \((u, v)\) of problem (1.1) is global and bounded. Furthermore, if
then we have the following decay estimates:
-
(i)
if \(p=q=1\), then, for \(t\geq0\),
$$E(t)\leq E(0)e^{-\tau_{1}t}, $$ -
(ii)
if \(\max\{p,q\}>1\), then, for \(t\geq0\),
$$E(t)\leq \biggl(E^{-\max\{\frac{p-1}{2},\frac{q-1}{2}\}}(0)+\tau_{2}\max \biggl\{ \frac{p-1}{2},\frac{q-1}{2} \biggr\} [t-1]^{+} \biggr)^{-\frac{2}{\max\{ p,q\}-1}}, $$
where \(\tau_{1}\) and \(\tau_{2}\) are some positive constants.
Proof
First, to prove that \(T=\infty\), it suffices to show that \(\|\Delta u\|_{2}^{2}+\|\Delta v\|_{2}^{2}+\|\nabla u_{t}\|_{2}^{2}+\| \nabla v_{t}\|_{2}^{2}\) is bounded independently of t. Thanks to (3.3), (3.4), and (3.6), we have
Therefore,
where \(\alpha_{2}\) is a positive constant depending only on m. Thus, we obtain the global existence.
We further derive the decay rate of the energy function for problem (1.1) by Nakao’s method [18]. For this purpose, we have to show that the energy function defined by (3.3) satisfies the hypothesis of Lemma 2.6. Integrating (3.4) over \([t,t+1]\), we have
where
By (3.13), (3.14), and the Hölder inequality, we observe that
where \(c_{1}(\Omega)=\operatorname{vol}(\Omega)^{\frac{p-1}{p+1}}\) and \(c_{2}(\Omega )=\operatorname{vol}(\Omega)^{\frac{q-1}{q+1}}\).
By the mean value theorem there exist \(t_{1}\in[t,t+\frac{1}{4}]\) and \(t_{2}\in[t+\frac{3}{4},t+1]\) such that
Next, multiplying Eq. (1.1)1 by u and Eq. (1.1)2 by v, integrating over \(\Omega\times[t_{1},t_{2}]\), and using integration by parts, we obtain
Integrating by parts and applying the Cauchy-Schwarz inequality in the first term of the right-hand side of (3.17), we obtain
and
Now, we estimate the third term of the right-hand side of inequality (3.17). By the Cauchy-Schwarz inequality we have
and
Since
and
Now, we will estimate the right-hand side of (3.26). First, by (3.16), (2.1), and (3.8), letting \(\beta=\min\{l,k\}\), we have
and
By the Hölder inequality and (3.13) we find
Then we have
and similarly we obtain
By (3.13) and (3.14) we observe that
By the mean value theorem there exist \(t_{1}\in[t,t+\frac{1}{4}]\) and \(t_{2}\in[t+\frac{3}{4},t+1]\) such that
and
Applying Young’s inequality to convolution \(\|\phi\ast\psi\|_{q}\leq\| \phi\|_{r}\|\psi\|_{s}\) with
and noting that if \(q=1\), then \(r=1\) and \(s=1\), we get
and
From (3.1), (3.9), (3.10), (3.35), (3.36), and (A1) we have
To estimate the eleventh and twelfth terms on the right-hand side of (3.26), we use (3.37) to obtain
Using Hölder inequality, (2.1), (3.8), and (3.13), we have
and
Therefore, applying (3.13)-(3.15), (3.27)-(3.31), (3.33)-(3.34), and (3.37)-(3.40) to (3.26) we obtain
where
Note that the assumption
gives \(\beta_{2}>0\). Thus,
where
On the other hand, from the definition of \(J(t)\) and \(I(t)\), (3.9), and (3.10) we have
Hence, integrating (3.3) over \((t_{1},t_{2})\) and then using (3.43), (3.42), and (3.15), we deduce that
where \(c_{5}:=c_{3}c_{4}\). By integrating (3.4) over \([t,t_{2}]\) we obtain
Since \(t_{2}-t_{1}\geq\frac{1}{2}\) and \(E(t)\) is nonincreasing in t, it follows that
Then, we have
Consequently, since \(E(t)\) is nonincreasing, combining (3.44) with (3.46), we obtain
Then a simple application of Young’s inequality gives, for \(t\geq0\),
where \(c_{6}\) and \(c_{7}\) are positive constants.
Therefore, we have the following decay estimate.
(i) For \(p=q=1\), from (3.47) and (3.12) we get
where we choose \(c_{8}>1\). Thus, by Lemma 2.6 we obtain
where \(\tau_{1}:=\ln\frac{c_{8}}{c_{8}-1}\).
(ii) For \(\max\{p, q\}>1\), it follows from (3.47) that, for \(t\geq0\),
Since \(D_{1}(t)\leq E^{\frac{1}{p+1}}(t)\leq E^{\frac{1}{p+1}}(0)\) and \(D_{2}(t)\leq E^{\frac{1}{q+1}}(t)\leq E^{\frac{1}{q+1}}(0)\) by (3.12) and (3.4), we have, for \(t\geq0\),
Then, setting \(\rho:=\max \{\frac{p-1}{2},\frac{q-1}{2} \}\), we obtain
where \(c_{10}:=2^{\rho}\cdot c_{9}^{1+\rho}\) and \(c_{11}:=c_{10}\max (E^{\frac{2\rho-p+1}{p+1}}(0),E^{\frac{2\rho-q+1}{q+1}}(0) )\). Application of Lemma 2.6 to (3.48) yields
with \(\tau_{2}:=c_{11}^{-1}\).
The proof of Theorem 3.4 is completed. □
4 Blow-up result
In this section, we deal with the blow-up of solution to problem (1.1) with \(\gamma=\delta=1\). In order to state our result, we make an extra assumption on \(g_{1}\) and \(g_{2}\),
where \(\lambda_{1}\) and \(E_{1}\) are given in (4.3) and (4.4), respectively.
Next, we define the functional G that helps in establishing desired results by
where η is the constant from Lemma 2.3.
Remark 4.1
-
(i)
We can verify that the functional G is increasing in \((0,\lambda_{1})\), decreasing in \((\lambda_{1},+\infty)\), \(G(\lambda)\rightarrow-\infty\) as \(\lambda\rightarrow+\infty\), and G has attains the maximum
$$ E_{1}:=G(\lambda_{1})=\frac{m-1}{2(m+1)} \lambda_{1}^{2} $$(4.3)at
$$ \lambda_{1}:= \biggl(\frac{1}{\eta(m+1)} \biggr)^{\frac{1}{m-1}}. $$(4.4) -
(ii)
We observe from (3.3), Lemma 2.3, and (4.3) that
$$\begin{aligned} E(t)&\geq J(t)\geq\frac{1}{2}w^{2}(t)- \int_{\Omega}F(u,v)\, dx \\ &\geq\frac{1}{2}w^{2}(t)-\eta\bigl(l\|\nabla u \|_{2}^{2} +k\|\nabla v\|_{2}^{2} \bigr)^{\frac{m+1}{2}} \\ &\geq\frac{1}{2}w^{2}(t)-\eta w^{m+1}(t)=G\bigl(w(t) \bigr), \end{aligned}$$(4.5)where
$$\begin{aligned} w(t) :=& \bigl(\|\Delta u\|_{2}^{2}+\|\Delta v \|_{2}^{2}+\|\nabla u_{t}\| _{2}^{2}+ \|\nabla v_{t}\|_{2}^{2} +l\|\nabla u \|_{2}^{2} \\ &{}+k\|\nabla v\|_{2}^{2}+g_{1} \circ\nabla u+g_{2}\circ\nabla v \bigr)^{\frac{1}{2}}. \end{aligned}$$(4.6)
Before we state and prove our main result, we need the following lemma, which is similar to a lemma from [17] to study some classes of the coupled equations.
Lemma 4.2
Assume that (A1) and (2.4) hold, \(u_{0},v_{0}\in H_{0}^{2}(\Omega)\), and \(u_{1},v_{1}\in H_{0}^{1}(\Omega)\). Let \((u,v)\) be a solution of (1.1) with initial data satisfying \(E(0)< E_{1}\) and \(w(0)>\lambda_{1}\), that is,
Then there exists \(\lambda_{2}>\lambda_{1}\) such that, for all \(t\geq0\),
Proof
The proof is similar to that of Lemma 4.2 in [17]. □
Theorem 4.3
Suppose that (A1), (2.4), and (4.1) hold, \(u_{0},v_{0}\in H_{0}^{2}(\Omega)\), and \(u_{1},v_{1}\in H_{0}^{1}(\Omega)\). Assume further that \(m>\max(p,q)\) and \(I(0)<0\). Suppose that one of the following is satisfied:
-
(i)
\(E(0)<0\),
-
(ii)
\(0\leq E(0)< E_{1}\) and \(w(0)>\lambda_{1}\).
Then the solution of problem (1.1) blows up at a finite time, that is, there exists \(T<+\infty\) such that
Proof
For case (ii), \(0\leq E(0)< E_{1}\), set
where \(E_{2}:=\frac{E_{1}+E(0)}{2}\). By (3.4) we see that \(H'(t)\geq 0\). Thus, we obtain
Moreover, from (4.5), (4.8), and (4.3) we see that
Then, by (4.11), (4.12), and Lemma 2.2 we have
Let
where ϵ and σ are positive constants to be specified latter. By taking the derivative of (4.14) and using Eq. (1.1) with \(\gamma=\delta=1\) we get
Using the Hölder and Young inequalities, we observe that
and
Taking (4.16) and (4.17) into account, using (4.10) and the definition of \(E(t)\) by (3.3) to substitute for \(\int_{\Omega }F(u,v)\,dx\), (4.15) becomes
where
By (4.1) we observe that \(a_{3}>0\), and then by the definition of \(w(t)\) by (4.6) we have
Since \(w(t)\geq\lambda_{2}\) by (4.8) and \(\lambda_{2}>\lambda_{1}\) by Lemma 4.2, we note that
where \(c_{2}= {a_{3}\frac{\lambda^{2}_{2}-\lambda^{2}_{1}}{\lambda ^{2}_{2}}>0}\) and \(c_{3}=a_{3}\lambda_{1}^{2}-(m+1)E_{2}\).
Furthermore, by the definition of \(E_{1}\), \(E_{2}=\frac{E_{1}+E(0)}{2}\) and assumption (4.1) we see that
Therefore, based on the above arguments, we conclude that
To proceed further, by the Hölder and Young inequalities we have
and
where \(\delta_{1}\) and \(\delta_{2}\) are positive constants depending on t and will be specified later.
Then, inserting the last two inequalities into (4.18), we obtain
At this point, choosing \(\delta_{1}\) and \(\delta_{2}\) such that
we get that
where \(M_{1}\), \(M_{2}\) are positive constants, and \(M=\frac {pM_{1}}{p+1}+\frac{qM_{2}}{q+1}\). It follows from (4.13) that
and
Substitution of these two inequalities into (4.19) yields
Since \(p< m\) and \(q< m\), we note that
where \(c_{4}=\operatorname{vol}(\Omega)^{\frac{m-p}{m+1}}\) and \(c_{5}=\operatorname{vol}(\Omega )^{\frac{m-q}{m+1}}\). Thus,
where the last inequality is derived from
because of
and the constants \(c_{7}=c_{6}c_{4}\) and \(c_{8}=c_{6}c_{5}\).
Now, letting
we have
Hence, by the inequality
we have
and
where \(c_{9}\) and \(c_{10}\) are positive constants. Taking (4.24) and (4.25) into consideration and using the definition of \(w(t)\), (4.20) takes the form
where \(c_{11}=c_{6}\cdot c_{7}\cdot c_{9}\) and \(c_{12}=c_{6}\cdot c_{8}\cdot c_{10}\).
At this moment, setting \(a_{4}=\min\{c_{2}\beta,\frac{m+1}{2}\}\), decomposing \(\epsilon(m+1)H(t)\) in (4.26) by
and using (4.10) and Lemma 2.2, we obtain
Choosing \(M_{1}\) and \(M_{2}\) large enough such that
we get
for some positive constants \(c_{i}\), \(i=13,14,\ldots,17\). Once \(M_{1}\) and \(M_{2}\) are fixed, we pick \(\epsilon>0\) small enough such that
and
Thus, there exists \(K>0\) such that
which, together with (4.27), implies that
On the other hand, by the Hölder and Young inequalities, (4.22), and (4.23) we have that
and, similarly,
By using (4.9) we get
and
Substitution of these two inequalities into (4.30) yields
Similarly, we obtain
By using (4.29), (4.33)-(4.34), and (4.13) we get, for \(t\geq0\),
where \(c_{i}\), \(i=18,19,\ldots,24\), are positive constants. Combining (4.28) and (4.35), we get
where \(c_{25}=\frac{\epsilon K}{c_{24}}\). Integration of (4.36) over \((0,t)\) then yields
This shows that \(A(t)\) blows up in finite time T and
Furthermore, we get from (4.35) that
Thus, the solution of problem (1.1) blows up in finite time.
For case (i), \(E(0)<0\), we set \(H(t)=-E(t)\) instead of (4.10). Then, applying the same arguments as in case (ii), we have the desired result. □
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Acknowledgements
The authors cordially thank the anonymous referee for his (her) valuable comments and suggestions, which lead to the improvement of this paper. This work was partially supported by NNSF of China (61374089), NSF of Shanxi Province (2014011005-2), Shanxi Scholarship council of China (2013-013), and Shanxi international science and technology cooperation projects (2014081026).
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Hao, J., Zhang, X. On the decay and blow-up of solution for a system of nonlinear viscoelastic plate equations with dissipative terms. Bound Value Probl 2016, 115 (2016). https://doi.org/10.1186/s13661-016-0622-7
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DOI: https://doi.org/10.1186/s13661-016-0622-7