Abstract
In this paper, we study the initial boundary value problem for the following viscoelastic wave equation with Balakrishnan–Taylor damping and a delay term where the relaxation function satisfies \(g'(t)\leq -\xi (t)g^{r}(t)\), \(t\geq 0\), \(1\leq r< \frac{3}{2}\). The main goal of this work is to study the global existence, general decay, and blow-up result. The global existence has been obtained by potential-well theory, the decay of solutions of energy has been established by introducing suitable energy and Lyapunov functionals, and a blow-up result has been obtained with negative initial energy.
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1 Introduction
In this paper, we consider the following initial-boundary value problem with a delay term
where \(\Omega \subset \mathbb{R}^{n}\) is a bounded domain with sufficiently smooth boundary ∂Ω. \(p\geq 4, a, b, \alpha , \mu _{1}\) are fixed positive constants, \(\mu _{2}\) is a real number, \(\tau >0\) represents the time delay, and g is a positive function.
In the absence of the Balakrishnan–Taylor damping \((\alpha =0)\), Problem (1.1) is reduced to the well-known nonlinear wave equation with \(b=g=0\) and a Kirchhof-type wave equation with \(g=0\), which has been extensively studied, see for instance [5, 8, 13, 24, 30, 31, 35, 38, 41, 42] and the references therein. Balakrishnan–Taylor damping \((\alpha \neq 0)\), \(g=0\), and \(\mu _{1}=\mu _{2}=0\), was initially proposed by Balakrishnan and Taylor [2], and Bass and Zes [3]. It is related to the panel flutter equation and to the spillover problem. So far, it has been studied by many authors, we refer the interested readers to [12, 15, 32, 39, 43, 44] and the references therein. Zarai and Tatar [44] studied the following problem
They proved the global existence and the polynomial decay of the problem. Exponential decay and blow up of the solution to the problem were established in Tatar and Zarai [39].
It is well known that time-delay effects often appear in many chemical, physical, and economical phenomena because these phenomena depend not only on the present state but also on the past history of the system. Nicaise and Pignotti [33] considered the following wave equation with a delay term
They obtained some stability results in the case \(0<\mu _{2}<\mu _{1}\). Then, they extended the result to the time-dependent delay case in the work of Nicaise and Pignotti [34]. Kirane and Said-Houari [23] considered a viscoelastic wave equation with time delay
They proved the global well posedness of solutions and established the decay rate of energy for \(0<\mu _{2}<\mu _{1}\). Kafini et al. [17] investigated the following nonlinear wave equation with delay
They proved the blow-up result of solutions with negative initial energy and \(p\geq m\), and we refer the interested readers to [9, 10, 18, 27] and the references therein. For the viscoelastic wave equation with Balakrishnan–Taylor damping and time delay, Lee et al. [25] studied the following equation
and established a general energy decay result by suitable Lyapunov functionals. Gheraibia et al. [14] considered the following equation
and proved the general decay result of the solution in the case \(|\mu _{2}|< \mu _{1}\). For the related works of PDEs with time delay, see for instance [6, 7, 11, 16, 19–22, 26, 28, 36, 37, 40] and the references therein.
Motivated by the previous work, in this paper, we consider the problem (1.1) and under suitable assumptions on the relaxation functions g, we prove the global existence, general decay and the finite-time blow-up results of the solutions.
The outline of this paper is as follows: In Sect. 2, we give some preliminary results. In Sect. 3, we obtain the global existence of the solution of (1.1). Section 4 and Sect. 5 cover the general decay and blow-up of solutions, respectively.
2 Some preliminaries
In this section, we give some notation for function spaces and preliminary lemmas. Denote by \(\|\cdot \|_{p}\) and \(\|\cdot \|_{H^{1}}\) to the usual \(L^{p}(\Omega )\) norm and \(H^{1}(\Omega )\) norm, respectively.
For the relaxation function g, we assume
\((A_{1})\): \(g:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) is a nonincreasing differentiable function satisfying
\((A_{2})\): There exist a nonincreasing differentiable function ξ with \(\xi (0)>0\) satisfying
\((A_{3})\): The constant p satisfies
\((A_{4})\): The constants \(\mu _{1}\) and \(\mu _{2}\) satisfy
Assume further that g satisfies
Lemma 2.1
(Sobolev–Poincare inequality [1]). Let q be a number with \(2\leq q<\infty \,(n=1,2)\) or \(2\leq q<\frac{2n}{n-2}\,(n\geq 3)\), then, there is a constant \(c_{*}=c_{*}(\Omega ,q)\) such that
By using direct calculations, we have
where
To deal with the time-delay term, motivated by Nicaise and Pignotti [33], we introduce a new variable
which gives us
Then, problem (1.1)is equivalent to
Let ζ be a positive constant satisfying
We first state a local existence theorem that can be established.
Theorem 2.2
Let \((A_{1})\)–\((A_{4})\) hold. Then, for every \((v_{0},v_{1})\in H^{1}_{0}(\Omega )\times L^{2}(\Omega )\), \(f_{0}\in L^{2}(( \Omega )\times (0,1))\), there exists a unique local solution of the problem (1.1) in the class
Now, we define the energy associated with problem (2.8) by
Lemma 2.3
Let \((v,z)\) be a solution of problem (2.8). Then,
Proof
Multiplying the first equation in (2.8) by \(v_{t}\), integrating over Ω, and using (2.5), we obtain
Multiplying the second equation in (2.8) by ζz and integrating over \(\Omega \times (0,1)\), we obtain
Using Young’s inequality, we have
Combining (2.12), (2.13), and (2.14), we obtain
where \(c_{0}=\min \{\mu _{1}-\frac{\zeta}{2\tau}-\frac{|\mu _{2}|}{2}, \frac{\zeta}{2\tau}-\frac{|\mu _{2}|}{2} \}\), which is positive by (2.9). The proof is complete. □
Next, we define the functionals
and
Then, it is obvious that
3 Global existence
In this section, we will prove that the global existence of the solution to (1.1) is in time.
Lemma 3.1
Assume that \((A_{1})\), \((A_{3})\)–\((A_{4})\) hold, and for any \((v_{0},v_{1})\in H^{1}_{0}(\Omega )\times L^{2}(\Omega )\), such that
then,
Proof
Since \(I(0) > 0\), then by the continuity of v, there exists a time \(T_{m}>0\) such that
From (2.16) and (2.17), we have
Thus, from \((A_{1})\), (2.11), (2.18), and (3.4), we obtain
Exploiting Lemma 2.1, (3.1), and (3.5), we obtain
Hence, we can obtain
By repeating the procedure, \(T_{m}\) is extended to T. The proof is complete. □
Theorem 3.2
Assume that the conditions of Lemma 3.1hold, then the solution (1.1) is global and bounded.
Proof
It suffices to show that \(\|v_{t}\|_{2}^{2}+\|\nabla v\|_{2}^{2}\) is bounded independently of t. By using (2.11), (2.18), and (3.5), we obtain
Therefore, we have
where \(K_{1}\) is a positive constant. □
4 General decay
In this section, we prove the general decay result by constructing a suitable Lyapunov functional.
Theorem 4.1
Let \((v_{0},v_{1})\in H_{0}^{1}(\Omega )\times L^{2}(\Omega )\). Assume that \((A_{1})\)–\((A_{4})\) hold. Then, there exist two positive constants K and k such that the solution of problem (1.1) satisfies, for all \(\forall t\geq t_{0}\),
Moreover, if
then
For this goal, we set
where ε is a positive constant to be specified later and
In order to show our stability result, we need the following lemmas:
Lemma 4.2
Let \((v,z)\) be a solution of problem (2.8). Then, there exist two positive constants \(\alpha _{1}\) and \(\alpha _{2}\) such that
for \(\varepsilon >0\) small enough.
Lemma 4.3
Assume that g satisfies \((A_{1})\) and \((A_{2})\), then
Corollary 4.4
([4]) Assume that g satisfies \((A_{1})\) and \((A_{2})\), and v is the solution of (1.1), then
Lemma 4.5
Let \((v,z)\) be a solution of problem (2.8). Then, the functional \(F(t)\) satisfies
where \(k_{1}\) and \(k_{2}\) are some positive constants.
Proof
Taking a derivation of (4.5), using (2.8), and Lemma 2.3, we obtain
By using Hölder’s, Young’s, Sobolev–Poincare inequalities, and \((A_{1})\), we obtain
and
and
Combining (4.10)–(4.12) and (4.9), we obtain
At this point, we choose η and ε so small that (4.7) remains valid and
Consequently, inequality (4.13) becomes
where \(k_{i},\,i=1,2\). are some positive constants. □
Now, we are ready to prove Theorem 4.1.
Proof of Theorem 4.1. Multiplying (4.14) by \(\xi (t)\), we obtain
4.1 Case: \(r=1\)
Using \((A_{2})\) and (2.11), then inequality (4.14) becomes
We choose \(G(t)=\xi (t)F(t)+2k_{2}E(t)\) that is equivalent to \(E(t)\) because of (4.7). Then, from (4.16) we can obtain
A simple integration of (4.17), leads to
which implies
4.2 Case: \(r>1\)
Applying Corollary 4.4, then inequality (4.15) becomes
Multiplying (4.20) by \(\xi ^{\nu}(t)E^{\nu}(t)\) where \(\nu =2r-2\), we have
Using Young’s inequality with \(q=\nu +1\) and \(q^{*}=\frac{\nu +1}{\nu}\), yields
At this point, we choose \(\eta <\frac{k_{1}}{k_{2}}\) and recall that \(\xi'(t)\leq0\) and \(E'(t)\leq0\), we obtain
which implies
We choose \(G(t)=\xi ^{\nu +1}(t)E^{\nu}(t)F(t)+k_{4}E(t)\) that is equivalent to \(E(t)\). Then,
A simple integration of (4.24) and using the fact that \(G(t)\sim E(t)\), leads to
4.3 Case: \(1< r<3/2\)
To establish (4.4), we note that from simple calculations show that (4.2) and (4.3) yield
Next, let
then, we have
Applying Jensens’s inequality for the second term on the right-hand side of (4.15) and using \((A_{2})\), we obtain
Multiplying (4.26) by \(\xi ^{\nu}(t)E^{\nu}(t)\), where \(\nu =r-1\), we have
The remainder of the proof is similar to (4.2). The proof is complete.
5 Blow up
In this section, we state and prove the blow up of the solution to problem (1.1) with negative initial energy.
Let
where \(E(0)<0\). From (5.1) and (2.11) we have
and \(H(t)\) is an increasing function. Using (2.10) and (5.1), we obtain
Moreover, similar to the work of Messaoudi [29], we can obtain the following lemma that is needed later.
Lemma 5.1
Suppose that \((A_{1})\), \((A_{3})\), \((A_{4})\), (2.4), and \(E(0)<0\) hold. Then, we have, for any \(2\leq s\leq p\),
where C is a positive constant.
Theorem 5.2
Let the conditions of Lemma 5.1hold. Then, the solution of problem (1.1) blows up in finite time.
Proof
Set
where \(\varepsilon >0\) is a small constant that will be chosen later, and
Taking a derivative of (5.4) and using the first equation in (2.8), we have
Applying Hölder’s and Young’s inequalities, for \(\eta ,\delta >0\), we have
and
Combining these estimates (5.7)–(5.9) and (5.6), we obtain
Applying (2.10) to the last term \(\|v\|_{p}^{p}\) on the right-hand side of (5.10) and using (5.1), we see that
for some number η with \(0<\eta <p/2\). By recalling (2.4), the estimate (5.11) reduces to
where
Therefore, by taking \(\delta =H(t)^{\sigma}/2c_{0}k\), where \(k>0\) is to be specified later, and exploiting (5.3), we se that
Substituting (5.13) into (5.12), we obtain
where \(c_{5}=(c_{p}^{2}(\mu _{1}^{2}+\mu _{2}^{2}))/2c_{0}p^{\sigma}\). From (5.5) and Lemma 5.1, for \(s=\sigma p+2\leq p\), we deduce
Combining (5.15) with (5.14), we obtain
Subtracting and adding \(\varepsilon \gamma H(t)\) on the right-hand side of (5.16), using (2.10) and (5.1), we deduce
First, we fix γ such that
Secondly, we take k large enough such that
Once k is fixed, we select \(\varepsilon >0\) small enough so that
Therefore, we obtain from (5.17) that
where ω is a positive constant.
We now estimate \(\Gamma (t)^{\frac{1}{1-\sigma}}\). By Hölder’s inequality, we have
which implies
Young’s inequality yields
for \(\frac{1}{\mu}+\frac{1}{\vartheta}=1\). To obtain \(\frac{\mu}{1-\sigma}= \frac{2}{1-2\sigma}\leq p\), by (5.5), we take \(\vartheta= 2(1-\sigma )\). Therefore, (5.21) becomes
where \(s=\frac{2}{1-2\sigma}\). Using Lemma 5.1, we obtain
Combining (5.4) and (5.22), we obtain
We note from (3.8) and (5.3) that
It follows from (5.23) and (5.24) that
Combining (5.25) with (5.18), we find that
A simple integration of (5.26) over \((0,t)\) yields
Consequently, the solution of problem (1.1) blows up in finite time \(T^{*}\) and \(T^{*}\leq \frac{1-\sigma}{\kappa \sigma \Gamma ^{\frac{\sigma}{1-\sigma}}(0)}\). □
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This work was supported by the Directorate-General for Scientific Research and Technological Development, Algeria (DGRSDT).
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Gheraibia, B., Boumaza, N. Initial boundary value problem for a viscoelastic wave equation with Balakrishnan–Taylor damping and a delay term: decay estimates and blow-up result. Bound Value Probl 2023, 93 (2023). https://doi.org/10.1186/s13661-023-01781-8
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DOI: https://doi.org/10.1186/s13661-023-01781-8