Exponential energy decay and blow-up of solutions for a system of nonlinear viscoelastic wave equations with strong damping
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In this paper, we consider the system of nonlinear viscoelastic equations
with initial and Dirichlet boundary conditions. We prove that, under suitable assumptions on the functions g i , f i (i = 1, 2) and certain initial data in the stable set, the decay rate of the solution energy is exponential. Conversely, for certain initial data in the unstable set, there are solutions with positive initial energy that blow up in finite time.
2000 Mathematics Subject Classifications: 35L05; 35L55; 35L70.
Keywordsdecay blow-up positive initial energy viscoelastic wave equations
where Ω is a bounded domain in ℝ n with a smooth boundary ∂Ω, and g i (·) : ℝ+ → ℝ+, f i (·, ·): ℝ2 → ℝ (i = 1, 2) are given functions to be specified later. Here, u and v denote the transverse displacements of waves. This problem arises in the theory of viscoelastic and describes the interaction of two scalar fields, we can refer to Cavalcanti et al. , Messaoudi and Tatar , Renardy et al. .
for b = 0 and b = 1 and for a wider class of relaxation functions. He established a more general decay result, for which the usual exponential and polynomial decay results are just special cases.
in Ω × (0, ∞) with initial and boundary conditions has extensively been studied. See in this regard, Kafini and Messaoudi , Messaoudi [11, 12], Song and Zhong , Wang . For instance, Messaoudi  studied (1.2) for h(u t ) = a|u t |m-2u t and f(u) = b|u|p-2u and proved a blow-up result for solutions with negative initial energy if p > m ≥ 2 and a global result for 2 ≤ p ≤ m. This result has been later improved by Messaoudi  to accommodate certain solutions with positive initial energy. Song and Zhong  considered (1.2) for h(u t ) = -Δu t and f(u) = |u|p-2u and proved a blow-up result for solutions with positive initial energy using the ideas of the "potential well'' theory introduced by Payne and Sattinger .
where Ω is a bounded domain with smooth boundary ∂Ω in ℝ n , γ1, γ2 ≥ 0 are constants and ρ is a real number such that 0 < ρ ≤ 2/(n - 2) if n ≥ 3 or ρ > 0 if n = 1, 2. Under suitable assumptions on the functions g(s), h(s), f(u, v), k(u, v), they used the perturbed energy method to show that the dissipations given by the viscoelastic terms are strong enough to ensure exponential or polynomial decay of the solutions energy, depending on the decay rate of the relaxation functions g(s) and h(s). For the problem (1.1) in ℝ n , we mention the work of Kafini and Messaoudi .
Motivated by the above research, we consider in this study the coupled system (1.1). We prove that, under suitable assumptions on the functions g i , f i (i = 1, 2) and certain initial data in the stable set, the decay rate of the solution energy is exponential. Conversely, for certain initial data in the unstable set, there are solutions with positive initial energy that blow up in finite time.
This article is organized as follows. In Section 2, we present some assumptions and definitions needed for this study. Section 3 is devoted to the proof of the uniform decay result. In Section 4, we prove the blow-up result.
as the usual L2(Ω) inner product.
For the relaxation functions g i (t) (i = 1, 2), we assume
We next state the local existence and the uniqueness of the solution of problem (1.1), whose proof can be found in Han and Wang  (Theorem 2.1) with slight modification, so we will omit its proof. In the proof, the authors adopted the technique of Agre and Rammaha  which consists of constructing approximations by the Faedo-Galerkin procedure without imposing the usual smallness conditions on the initial data to handle the source terms. Unfortunately, due to the strong nonlinearities on f1 and f2, the techniques used by Han and Wang  and Agre and Rammaha  allowed them to prove the local existence result only for n ≤ 3. We note that the local existence result in the case of n > 3 is still open. For related results, we also refer the reader to Said-Houari and Messaoudi  and Messaoudi and Said-Houari . So throughout this article, we have assumed that n ≤ 3.
3. Global existence and energy decay
In this section, we deal with the uniform exponential decay of the energy for system (1.1) by using the perturbed energy method. Before we state and prove our main result, we need the following lemmas.
Proof. The proof is almost the same that of Said-Houari , so we omit it here. □
where η is the optimal constant in (3.3). The following lemma will play an essential role in the proof of our main result, and it is similar to a lemma used first by Vitillaro , to study a class of a single wave equation, which introduces a potential well.
This is impossible since E(t) ≤ E(0) < E1 for all t ∈ [0, T). Hence (3.6) is established. □
The following integral inequality plays an important role in our proof of the energy decay of the solutions to problem (1.1).
for every t ≥ c.
for every t ≥ aC-1, where C is some positive constant.
where C is a positive constant depending only on p.
where C is a constant independent on u.
for every t ∈ [0, ∞).
for every t ≥ Ca-1. □
4. Blow-up of solution
From the assumption (G 2), we have θ i > 0 (i = 1, 2). Similarly Lemma 3.2, we have
where the constants θ i (i = 1, 2) are defined in (4.1).
Then we have
where α* is given in (4.3). Since E(0) < E2, there exists α2 > α* such that g(α2) = E(0).
This is impossible since E(t) ≤ E(0) for all t ∈ [0, T). Hence (4.5) is established. □
blows up in finite time, where the constants θ i (i = 1, 2) are defined in (4.1) and α*, E2are defined in (4.3).