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Global existence and blow-up of solutions for a nonlinear wave equation with memory
Journal of Inequalities and Applications volume 2012, Article number: 33 (2012)
Abstract
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.
1. Introduction
In this article we study the behavior of solutions for the following nonlinear viscoelastic equation
where Ω is a bounded domain in ℝnwith a smooth boundary ∂Ω, g is a positive function satisfying some conditions to be specified later, ω, μ satisfy
λ being the first eigenvalue of the operator -Δ under homogeneous Dirichlet boundary conditions, and
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 [5] 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 [11] 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.
This type of problem have been considered by many authors and several results concerning existence, nonexistence, and asymptotic behavior have been established. Cavalcanti et al. [17] studied the following equation:
for a : Ω → ℝ+, a function, which may be null on a part of the domain Ω. Under the conditions that a(x) ≥ a0 > 0 on Ω1 ⊂ Ω, with Ω1 satisfying some geometry restrictions and
the authors established an exponential rate of decay. This latter result has been improved by Cavalcanti and Oquendo [18] and Berrimi and Messaoudi [19]. In their work, Cavalcanti and Oquendo [18] considered the situation where the internal dissipation acts on a part of Ω and the viscoelastic dissipation acts on the other part. They established both exponential and polynomial decay results under conditions on g and its derivatives up to the third order, whereas Berrimi and Messaoudi [19] allowed the internal dissipation to be nonlinear. They also showed that the dissipation induced by the integral term is strong enough to stabilize the system and established an exponential decay for the solution energy provided that g satisfies a relation of the form
In [20], 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.
For nonexistence, we should mention that Messaoudi [21] looked into the equation
and proved, under appropriate relations between p, m and g, a blow-up result. This work generalizes earlier ones by Georgiev and Todorova [22] and Messaoudi [23], in which a similar result has been established for the wave equation (g ≡ 0). This result was later improved by Messaoudi [24], to certain solutions with positive initial energy. A similar result was also obtained by Wu [25] using a different method. For the problem (1.4) in ℝnand with m = 2, Kafini and Messaoudi [26] showed, for suitable conditions on g and initial data, that solutions with negative energy blow up in finite time. More recently, Wang [27] 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 [28], 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.
2. Preliminaries
We denote by ∥ · ∥ q the Lq(Ω) norm for 1 ≤ q ≤ ∞ and by ∥∇ · ∥2 the Dirichlet norm in . Moreover, for later use we denote by 〈·,·〉 the duality pairing between H-1(Ω) and . When ω > 0 (resp. ω = 0) for v, (resp. for all v, w ∈ L2(Ω)), we put
by (1.2), ∥ · ∥* is an equivalent norm over (resp. L2(Ω)).
Let a > 0. Define J a , I a : by
In this case, the "potential depth" is defined as
It is easy to see that the "potential well" is positive, see [28, 29] for details. Next, we define stable and unstable sets respectively:
Finally, we consider the energy functional E(t) = E(u(t),u t (t)) defined by
where
For the relaxation function g(t) we assume
(G1) g ∈ C1[0, ∞) is a non-negative and non-increasing function satisfying
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).
For a given T > 0, we consider the space equipped with the norm
Lemma 3.1. Assume (G 1), (1.2) and (1.3) hold. For every T > 0, every and every initial data there exists a unique
which solves the linear problem
Proof. The proof follows from a directly application of the Galerkin method as in [22, 30], thus we omit it here.
Theorem 3.2. Assume (G 1), (1.2) and (1.3) hold. For any initial data , there exists a real number T m > 0 such that problem (1.1) has a unique local weak solution
If T m < ∞, then
Proof. Taking and letting . For any T > 0, we consider
By Lemma 3.1, for any we may define v = Φ(u), being the unique solution to problem (3.1). We claim that, for a suitable T > 0, Φ is a contractive map from into itself. Given , multiplying (3.1) by v t and integrating over [0,t] ⊂ [0,T], we have
here taking into account the condition (G 1). For the last term, using Hölder, Sobolev, and Young inequalities, we have
where 2* = 2n/(n-2). Combining (3.3) with (3.4) and taking the maximum over [0, T], we get
Choosing T sufficiently small such that C2TR2(p-1)≤ R2/2, we get , which shows that Φ maps into itself.
Next, we verify that Φ is a contraction. Taking w1 and w2 in , subtracting the two equations (3.1) for v1 = Φ(w1) and v2 = Φ(w2) and setting w = v1- v2, then we have for all and a.e. t ∈ [0,T]
By taking φ = w t in (3.5) and arguing as above, we obtain
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 ).
By the construction above, we observe that the local existence time of u merely depends (through R) on the norms of the initial data. Therefore, as long as remains bounded, the solution may be continued, see also [[31], p. 158], for a similar argument. Hence, if T m < ∞, we have
Before we state and prove our global existence result, we need the following lemmas.
Lemma 3.3. [24, Lemma 2.1] Assume (G 1), (1.2) and (1.3) hold. Let u(t) be a solution of (1.1). Then E(t) is nonincreasing, that is
Moreover, the following energy inequality holds:
Lemma 3.4. Assume (G 1), (1.2) and (1.3) hold, and 0 < a ≤ l. Let u(x, t) be a local solution of problem (1.1) with initial data . Then the following assertions hold.
-
(1)
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 ).
-
(2)
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 [32].
The following integral inequality plays an important role in our proof of the energy decay of the solutions to problem (1.1).
Lemma 3.5. [33] Assume that the function φ : ℝ+ ∪ {0} → ℝ+ ∪ {0} is a non-increasing function and that there exists a constant c > 0 such that
for every t ∈ [0, ∞). Then
for every t ≥ c.
Theorem 3.6. Assume (G 1), (1.2) and (1.3) hold, and 0 < a ≤ l. Let u(x, t) be a local solution of problem (1.1) with initial data . In addition assume that u(0) ∈ W a and E(0) < d a , then the corresponding solution to (1.1) globally exists, i.e., T m = ∞. Moreover, if d a < θ and σ = 1 - l > 0 is small sufficiently such that
where and C(Ω) is the optimal constant of Sobolev imbedding , then the energy decay is
for every t ∈ [0, ∞), where C is some positive constant.
Proof. We only consider the case ω > 0 and μ > - λω. In order to get T m = ∞, by Theorem 3.2, it suffices to show that
is bounded independently of t. Since u(0) ∈ W a and E(0) < d a , it follows from Lemma 3.4 that
On the other hand, since u(t, ·) ∈ W a means
So, it follows from (3.8) and Lemma 3.3 with s = 0 that
which implies
where C is a positive constant depending only on l and p.
From Lemma 3.3 we have
which together with u(t, ·) ∈ W a yields
In addition,
where
Note that E(0) < θ, we see that ϵ > 0.
Multiplying (1.1) by u(t) and integrating over Ω × [t1,t2] (0 ≤ t1 ≤ t2), we get
For the last term in (3.11), one has
Combining (3.11) and (3.12), we have
where the last inequality comes from (G 1). For the left-hand side of the (3.13), by (3.10) we obtain
We next estimate every term of the right-hand side of the (3.13). Firstly, by Hölder inequality and Poincaré inequality
Using (3.9) we see that
where c1 is a constant independent on u, from which follows that
Since u(t, ·) ∈ W a , we have 0 < I a (u) ≤ E(t). Thus, from (3.7), we deduce that
which implies
Hence, by Poincaré inequality we get
where c3 is a constant independent on u. In addition, using Young's inequality for convolution ∥f*g∥ q ≤ ∥f∥ r ∥g∥ s with 1/q = 1/r + 1/s - 1 and 1 ≤ q,r,s ≤ ∞, noting that if q = 1, then r = 1 and s = 1, we have
Further, by (3.9) we then have
and
Combining (3.17) and (3.18), we get
By Poincaré inequality and (3.9), we also have the following estimate
where c4 is a constant independent on u.
Combining (3.13)-(3.20), we obtain
where C is a constant independent on u, that is
Denote
We rewrite (3.21)
for every t ∈ [0, ∞).
Since a > 0 when σ = 1 - l > 0 small sufficiently by Lemma 3.5, we obtain the following energy decay for problem (1.1) as
for every t ≥ Ca-1.
Remark 3.1. For the definition of d a and Sobolev imbedding inequality, we have
and
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 [34]. 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 [30], however our argument contains nontrival modifications.
Lemma 4.1. Assume (G 1), (1.2) and (1.3) hold. Let u(x,t) be a local solution of problem (1.1) on [0,T m ) with initial data . If T m < ∞, then
Moreover, if n ≥ 3 and p = 2n/(n - 2) = 2* (ω > 0), then (4.1) also holds for q = p.
Proof. From (3.7), we have
which, together with (3.2), implies
This proves (4.1) at once for the case of p = q = 2n/(n - 2). For the remaining cases, notice (4.3) that implies
Moveover, by (4.2) we obtain
From the Gagliardo-Nirenberg inequality we have
which yields
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).
Next we will prove the main blow-up result by the concavity method of Levine [35, 36] and the estimates similar as [30].
Theorem 4.2. Assume (G 1), (G 2), (1.2) and (1.3) hold. Let u(x,t) be a local solution of problem (1.1) with initial data . If ω > 0, then there is a real number t0 ∈ [0,T m ) such that u(t0, ·) ∈ V k and E(t0) < d k if and only if T m < ∞, where
Proof. We first consider "if part", without loss of generality, we may assume that t0 = 0. Assume by contradiction that the solution u is global. Then, for any T > 0 we consider H(t) : [0,T] → ℝ+ defined by
A direct computation yields
and
By multiplying (1.1) by u and integrating over Ω, we have
which implies
Therefore, we have
where G(t) : [0,T] → ℝ+ is the function defined by
Using the Schwarz inequality, we have
and
These three inequalities entail G(t) ≥ 0 for every [0, T]. Using (4.6), we get
where
For the last term on the left of (4.8), we have
Combining (4.8) with (4.9), we get
Using (3.7), we have
and then
Since
we have
By Lemma 3.4, we have
Then, we have
The above inequality comes from [29]; see [28, 29] for further details. Since E(0) < d k , there exists δ > 0 (independent of T) such that
From (4.10) and the definition of H(t), there also exists ρ > 0 (independent of T) such that
By (4.7), (4.11), and (4.12) it follows that
Setting y(t) = H(t)-(p-2)/4, then we have
which implies that y(t) reaches 0 in finite time, say as t → T*. Since T* is independent of the initial choice of T, we may assume that T* < T. This tells us that
In turn, this implies that
Indeed, if as t → T*, then (4.13) immediately follows. On the contrary, if remains bounded on [0,T*), then
so that again (4.13) is satisfied. This implies a contradiction, i,e., T m < ∞.
Conversely, for "only if part" we assume now that T m < ∞. Notice first that, for every t > 0, there holds
Hence, by (3.7) and 0 < k ≤ l, we have
By Lemma 4.1 we have ∥∇u(t)∥2 → ∞ as t → T m , i.e., ∥u(t)∥* → ∞ as t → T m , together with (4.14) which implies
Since J k (u(t)) ≤ E(t), by (4.15) we obtain that
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 [37] 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:
(G3) The function of is of positive in the following sense:
∀v ∈ C1([0,∞)) and ∀t > 0.
Obviously, g(t) = εe-twith 0 < ε < 1 satisfies assumptions (G1)-(G3). Let
Lemma 4.3. [27, Lemma 2.1]) Assume that g(t) satisfies (G 1), (G 3) and Λ(t) is a function that is twice continuously differentiable, satisfying
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 ).
Lemma 4.4. Suppose that , u1 ∈ L2(Ω) satisfy
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 ).
Proof. Since u(t) is the local solution of problem (1.1) with weak damping, by a simple computation we have
where the last inequality uses I(u(t)) < 0. Then we get
Therefore, this lemma comes from Lemma 4.3.
Theorem 4.5. Assume (G 1), (G 3), (1.2) and (1.3) hold. Let u(x,t) be a local solution of problem (1.1) with initial data . If ω = 0, g(s) also satisfies
and (u0,u1) satisfies the following conditions
where
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 < ∞.
Proof. Without loss of generality, we may assume μ = 1. First, by a contradiction argument we claim that
and
for every t ∈ [0,T m ). If this was not the case, then there would exist a time t1 such that
By the continuity of the solution u(t) as a function of t, we see that I(u(t)) < 0 when t ∈ (0,t1) and I(u(t)) = 0. Thus by Lemma 4.4 we have
for every t ∈ [0,t1). In addition, it is obvious that is continuous on [0,t1]. Thus the following inequality is obtained:
On the other hand, it follows from the definition of E(t) and (3.7) that
Since , from (4.22), we have
Noting the fact that I(u(t1)) = 0, we then have
Thus, by the Poincaré inequality and (4.16) we have
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 ).
Secondly, we prove that the solution of problem (1.1) blows up in a finite time. The proof is similar "if part" in the Theorem 4.2, for the convenience of the readers, we give the sketch of the proof here. Assume by contradiction that the solution u is global. Then, for sufficiently large T > 0 we consider Φ(t) : [0,T] → ℝ+ defined by
A direct computation yields
and
where (u(t),u t (t)) = ∫Ωu(t)u t (t)dx. By multiplying (1.1) by u and integrating over Ω, we have
which implies
Therefore, we have
where Ψ(t) : [0,T] → ℝ+ is the function defined by
Using the Schwarz inequality, we have
and
These three inequalities entail Ψ(t) ≥ 0 for every [0,T]. Using (4.24), we get
where
Combining (4.9) with (4.26), we get
Using (3.7) for ω = 0, we have
and then
where the last inequality follows from Lemma 4.4 and the Poincaré inequality. Since 0 < k < 1, we have pk - 2 < (p - 2)k. From (4.17), we get
Therefore, there exists δ1 > 0 (independent of T) such that
From Lemma 4.4, (4.17) and the definition of Φ(t), there also exists ρ1 > 0 (independent of T) such that
By (4.24), (4.27), and (4.28) it follows that
The rest of the proof is the same as "if part" in the Theorem 4.2, so we omit it here.
5. The boundedness of global solution
In this section, we will prove the boundedness of global solutions u(t) to problem (1.1) for strong (ω > 0) damping, namely,
Throughout this section, we assume that
If (5.2) holds, then the solution to problem (1.1) for strong (ω > 0) damping is global. Indeed, if u(t) blows up in finite time, by Theorem 4.2, E(t0) < d k for some t0 > 0. Hence, E(u(t),u t (t)) = E(t) < d k for all t ≥ t0. This is a contradiction.
Since for a.e. t ≥ 0, we combine Poincare inequality with (3.7) and (5.2) to show that, for every t > 0 we have
Letting t → ∞, we conclude that
Furthermore, observe that by the definition of E(t), we have
Since
from (4.5), we have
Combining (4.9) with (5.5), we get
where the last inequality follows from (5.4).
Inspired by Gazzola and Weth [38] we now prove a crucial stability result.
Lemma 5.1. Assume (G 1), (G 2), (1.2) and (1.3) hold. If u(t) is a solution to problem (1.1) for strong (ω > 0) damping satisfying E(u(t),u t (t)) = E(t) ≥ d k for all t ≥ 0, then we have
Proof. Fixed η > 0, by (3.7), for every t > 0 we have
Since E(t) is nonincreasing and lower bounded by d k , E(t) admits finite limit as t → ∞. This immediately yields the assertion by letting t → ∞ in the previous inequality.
Theorem 5.2. Assume (G1), (1.2) and (1.3) hold. In addition, g(s) also satisfies
If u(t) is a solution to problem (1.1) for strong (ω > 0) damping satisfying E(u(t),u t (t)) = E(t) ≥ d k for all t ≥ 0, then the solution u(t) satisfies (5.1).
Proof. Assuming by contradiction that (5.1) fails, namely that there exists a diverging sequence t j ⊂ ℝ+ such that
Then, by the definition of E(t) and (5.2), we have ∥u(t j )∥ p → ∞ as j → ∞. By Sobolev inequality we get
By (5.9) and continuity, we can select a diverging sequence such that . Moreover, by Lemma 5.1, we have
Then, we find a second diverging sequence τ m ⊂ ℝ+ such that
In view of (5.3), for all m sufficiently large,
Clearly, up to renaming τ m into (τ m - 1) we now have
Also, for m large enough, there holds
Indeed, by (5.10), (5.12), Young, Hölder, and Poincaré inequalities,
for every m large enough. By (5.13) integrating (5.6) on the time interval [t m ,t] for t ∈ (t m ,t m + τ m ] entails
provided m is sufficiently large, where the last inequality follows from (5.7) and the equivalent norm ∥·∥* and . On the other hand, by Young, Hölder, and Poincaré inequalities,
Set
Combining (5.14) with (5.15), we have the following differential inequality
for some γ > 0 and c3 > 0. Hence
By (5.12), we have
Then, from (5.16) and (5.17), we obtain
Integrating (5.18) over and taking into account (5.3) we find
where we have set . Hence, up to enlarging m, we may take the exponential and we finally conclude that
where we also used (5.17). On the other hand, by inequality (5.12), it turns out that
which contradicts (5.19) as τ m → ∞. Therefor, (5.8) is false and {u(t)} is bounded, namely there exists C such that
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Acknowledgements
The authors were indebted to the referee for giving some important suggestions which improved the presentations of this article. Supported in part by a China NSF Grant No. 10871097, Qing Lan Project of Jiangsu Province, the NSF of the Jiangsu Higher Education Committee of China (11KJA110001), the Foundation for Young Talents in College of Anhui Province Grant No. 2011SQRL115, Program sponsored for scientific innovation research of college graduate in Jangsu province No. 181200000649, the pre-research project of Anhui Science and Technology University No. ZRC2012308 and the courses building projects of Anhui Science And Technology University No. ZDKC1121.
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FL and HG carried out all studies in this article. All authors read and approved the final manuscript.
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Liang, F., Gao, H. Global existence and blow-up of solutions for a nonlinear wave equation with memory. J Inequal Appl 2012, 33 (2012). https://doi.org/10.1186/1029-242X-2012-33
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DOI: https://doi.org/10.1186/1029-242X-2012-33