Global existence and blow-up of solutions for a nonlinear wave equation with memory

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.

In this article we study the behavior of solutions for the following nonlinear viscoelastic equation

where Ω is a bounded domain in ℝⁿ with 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 ℝⁿ and 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.

We denote by ∥ · ∥_q the L^q(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).

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 (G1), (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 (G1), (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 (G1). 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 C₂TR^2(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 (G1), (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 (G1), (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 (G1), (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 (G1). 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.

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 (G1), (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 (G1), (G2), (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^-t with 0 < ε < 1 satisfies assumptions (G1)-(G3). Let

Lemma 4.3. [27, Lemma 2.1]) Assume that g(t) satisfies (G1), (G3) 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 (G1), (G3), (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.

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 (G1), (G2), (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

The authors declare that they have no competing interests.

FL and HG carried out all studies in this article. All authors read and approved the final manuscript.

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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