Global existence and blow-up of solutions for a nonlinear wave equation with memory
© Liang and Gao; licensee Springer. 2012
Received: 12 June 2011
Accepted: 15 February 2012
Published: 15 February 2012
In this article, we consider the nonlinear viscoelastic equation
with initial conditions and Dirichlet boundary conditions. We first prove a local existence theorem and show, for some appropriate assumption on g and the initial data, that this solution is global with energy which decays exponentially under the potential well. Secondly, not only finite time blow-up for solutions starting in the unstable set is proved, but also under some appropriate assumptions on g and the initial data, a blow-up result with positive initial energy is established. Finally, we also prove the boundedness of global solutions for strong (ω > 0) damping case.
2000 MSC: 35L05; 35L15; 35L70.
This problem has its origin in the mathematical description of viscoelastic materials. It is well known that viscoelastic materials exhibit natural damping, which is due to the special property of these materials to retain a memory of their past history. A general theory concerning problem (1.1) in the case ω = 0 and μ = 0 is available in literature (see [1–4]). The asymptotic behavior of the solutions to (1.1) has been studied in [5–8], we also refer to [9, 10] for the asymptotic decay of the solutions to problems analogous to (1.1). Among other known results about problem (1.1) with ω = 0 and μ = 0, we recall that in [7, 8], it is proved that the exponential decay of g is a sufficient condition to the exponential decay of the solution u. In  it is also proved that, when ω = 0 and μ = 0, the exponential decay of g is necessary for the exponential decay of u. When ω + μ ≠ 0, Fabrizio and Polidoro  showed that the exponential decay of g is a necessary condition for the exponential decay of u. The case of only having may be very restrictive in many physical problems. Also, problem (1.1) is applied to the theory of the heat conduction with memory, see [12–16]. Therefore, the dynamics of (1.1) are of great importance and interest as they have wide applications in natural sciences.
In , Berrimi and Messaoudi considered problem (1.1) for ω = μ = 0. They established a local existence result and showed, for certain initial data and suitable conditions on g, that this solution is global with energy which decays exponentially or polynomially depending on the rate of the decay of the relaxation function g.
and proved, under appropriate relations between p, m and g, a blow-up result. This work generalizes earlier ones by Georgiev and Todorova  and Messaoudi , in which a similar result has been established for the wave equation (g ≡ 0). This result was later improved by Messaoudi , to certain solutions with positive initial energy. A similar result was also obtained by Wu  using a different method. For the problem (1.4) in ℝ n and with m = 2, Kafini and Messaoudi  showed, for suitable conditions on g and initial data, that solutions with negative energy blow up in finite time. More recently, Wang  has investigated a sufficient condition of the initial data with arbitrarily positive initial energy such that the corresponding solution of Equation (1.4) with m = 2 blows up in finite time. This result improved the blow-up results in [21, 24].
In this article, we first consider (1.1) and establish a local existence result. In addition, using the ideas of the "potential well" theory introduced by Payne and Sattinger , we show that for some appropriate assumption on g (but without exponential decay property) and the initial data, that this solution is global with energy which decays exponentially under the potential well. Secondly, not only finite time blow up for solutions starting in the unstable set is proved, but also under some appropriate assumptions on g and the initial data, a blow-up result with positive initial energy is established. Finally, we also prove the boundedness of global solutions u(t) to problem (1.1) for strong (ω > 0) damping, namely, .
This article is organized as follows. In Section 2 we introduce some notation and prepare some material. Section 3 is devoted to global existence for solutions under the potential well and the decay result. In Section 4 we will show that there are solutions of (1.1) with positive initial energy or with arbitrary positive initial energy that blow up in finite time. The last Section we will prove the boundedness of global solutions u(t) to problem (1.1) for strong (ω > 0) damping.
by (1.2), ∥ · ∥* is an equivalent norm over (resp. L2(Ω)).
For the relaxation function g(t) we assume
Remark 2.1. Condition (1.3) is needed to establish the local existence result. In fact under this condition, the nonlinearity is Lipschitz from H1(Ω) to L2(Ω). Condition (G1) is necessary to guarantee the hyperbolicity and well-posedness of problem (1.1).
3. Global existence and exponential energy decay
In this section we study the global existence of solutions for problem (1.1). For this purpose, we first consider a related linear problem. Then, we use the well-known contraction mapping theorem to prove the existence of solutions to the nonlinear problem. Throughout the section, we restrict ourselves to the case ω > 0, μ ≠ 0 and n ≥ 3, the other cases being similar (and simpler).
Choosing T sufficiently small such that C2TR2(p-1)≤ R2/2, we get , which shows that Φ maps into itself.
for some ε < 1 provided T is sufficiently small. This proves the claim. By the contraction mapping principle, there exists a unique (weak) solution to (1.1) defined on [0,T m ).
Before we state and prove our global existence result, we need the following lemmas.
If there exists a number t0 ∈ [0,T m ) such that u(·,t0) ∈ W a and E(t0) < d a , then u(·, t) ∈ W a and E(t) < d a for all t ∈ [t0,T m ).
If there exists a number t0 ∈ [0,T m ) such that u(·, t0) ∈ V a and E(t0) < d a , then u(·,t) ∈ V a and E(t) < d a for all t ∈ [t0,T m ).
Proof. The proof is almost the same that of Tsutsumi .
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 ∈ [0, ∞), where C is some positive constant.
where C is a positive constant depending only on l and p.
Note that E(0) < θ, we see that ϵ > 0.
where c4 is a constant independent on u.
for every t ∈ [0, ∞).
for every t ≥ Ca-1.
Since is compact, the best constants and the best function v(x) in the above Sobolev imbedding inequality can be attained. For example, n = 1, p = ∞, Ω = (c, d) ⊂ ℝ, the best C and the best function v(x) are attained, see . In this case, . Then, we can take the initial data u0 = v(x) which yields the set of the initial data that yields the exponential decay is not empty.
4. Blow-up solution
In this section, we deal with the blow-up solutions of problem (1.1). The basic idea comes from , however our argument contains nontrival modifications.
Moreover, if n ≥ 3 and p = 2n/(n - 2) = 2* (ω > 0), then (4.1) also holds for q = p.
Since n(p - 2)/2 < q < p implies 0 < α < 1 and pα < 2, the above inequality combined with (4.4) immediately yields (4.1).
so that again (4.13) is satisfied. This implies a contradiction, i,e., T m < ∞.
for some t0 ∈ [0,T m ). These imply u(t0) ∈ V k , E(t0) < d k .
Remark 4.1. The "if part" of Theorem 4.2 means that the solution to (1.1) blows up in a finite time for suitable "large" initial data u0 and u1 in the sense of u0 ∈ V k and E(0) < d k . Also, (4.15) is an essential behavior for which the solution of (1.1) blows up in a finite time.
Remark 4.2. In Theorem 4.2, we restrict ω > 0 in order to prove the "only if part". In fact, if ω > 0, it is easy to obtain ∥u(t)∥* → ∞ as t → T m from ∥∇u(t)∥2 → ∞ as t → T m , which implies E(t) → -∞ as t → T m . If ω = 0 (only with weak damping), assuming 2 < p ≤ 2 + 2/n, then we can obtain ∥u (t)∥2 → ∞ as t → T m (see  for details) which yields that Theorem 4.2 also holds for the case of ω = 0 with 2 < p ≤ 2 + 2/n.
Next, we consider the blow-up solution of problem (1.1) for the case of weak damping (ω = 0) with arbitrary positive initial energy. We need an addition assumption on the relaxation function g:
∀v ∈ C1([0,∞)) and ∀t > 0.
for every t ∈ [0,T m ), where u(t) is the corresponding solution of problem (1.1) with weak damping. Then the function Λ(t) is strictly increasing on [0,T m ).
If the local solution u(t) of problem (1.1) with weak damping exists on [0,T m ) and satisfies I(u(t)) < 0, then is strictly increasing on [0,T m ).
Therefore, this lemma comes from Lemma 4.3.
and λ is the first eigenvalue of the operator -Δ under homogeneous Dirichlet boundary conditions, then the corresponding solution u(t) of problem (1.1) blows up in a finite time, i.e., T m < ∞.
Obviously, there is a contradiction between (4.21) and (4.23). Thus, we have proved that (4.18) is true for every for every t ∈ [0,T m ). Furthermore, by Lemma 4.4 we see that (4.19) is also valid on t ∈ [0,T m ).