A blow up result for viscoelastic equations with arbitrary positive initial energy

Abstract

In this paper, we consider the following viscoelastic equations

with initial condition and zero Dirichlet boundary condition. Using the concavity method, we obtained sufficient conditions on the initial data with arbitrarily high energy such that the solution blows up in finite time.

1 Introduction

In this work, we study the following wave equations with nonlinear viscoelastic term

(1.1)

where Ω is a bounded domain of Rⁿ with smooth boundary ∂Ω, p > 1, q > 1 and g is a positive function. The wave equations (1.1) appear in applications in various areas of mathematical physics (see [1]).

If the equations in (1.1) have not the viscoelastic term , the equations are known as the wave equation. In this case, the equations have been extensively studied by many people. We observe that the wave equation subject to nonlinear boundary damping has been investigated by the authors Cavalcanti et al. [2, 3] and Vitillaro [4, 5]. It is important to mention other papers in connection with viscoelastic effects such as Aassila et al. [6, 7] and Cavalcanti et al. [8]. Furthermore, related to blow up of the solutions of equations with nonlinear damping and source terms acting in the domain we can cite the work of Alves and Cavalcanti [9], Cavalcanti and Domingos Cavalcanti [10]. As regards non-existence of a global solution, Levine [11] firstly showed that the solutions with negative initial energy are non-global for some abstract wave equation with linear damping. Later Levine and Serrin [12] studied blow-up of a class of more generalized abstract wave equations. Then Pucci and Serrin [13] claimed that the solution blows up in finite time with positive initial energy which is appropriately bounded. In [14] Levine and Todorova proved that there exist some initial data with arbitrary positive initial energy such that the corresponding solution to the wave equations blows up in finite time. Then Todorova and Vitillaro [15] improved the blow-up result above. However, they did not give a sufficient condition for the initial data such that the corresponding solution blows up in finite time with arbitrary positive initial energy. Recently, for problem (1.1) with g ≡ 0 and m = 1, Gazzalo and Squassina [16] established the condition for initial data with arbitrary positive initial energy such that the corresponding solution blows up in finite time. Zeng et al. [17] studied blowup of solutions for the Kirchhoff type equation with arbitrary positive initial energy.

Now we return to the problem (1.1) with g ≢ 0; in [18] Cavalcanti et al. first studied

and obtained an exponential decay rate of the solution under some assumption on g(s) and a(x). At this point it is important to mention some papers in connection with viscoelastic effects, among them, Alves and Cavalcanti [9], Aassila et al. [7], Cavalcanti and Oquendo [19] and references therein. Then Messaoudi [20] obtained the global existence of solutions for the viscoelastic equation, at same time he also obtained a blow-up result with negative energy. Furthermore, he improved his blow-up result in [21]. Recently, Wang and Wang [22] investigated the following problem

and showed the global existence of the solutions if the initial data are small enough. Moreover, they derived decay estimate for the energy functional. In [23] Wang established the blow-up result for the above problem when the initial energy is high.

In this paper, motivated by the work of [23] and employing the so called concavity argument which was first introduced by Levine (see [11, 24]), our main purpose is to establish some sufficient conditions for initial data with arbitrary positive initial energy such that the corresponding solution of (1.1) blows up in finite time. To this, we first rewrite the problem (1.1) to the following equivalent form

(1.2)

where

We next state some assumptions on g(s) and real numbers p > 1, q > 1.

(A1) g ∈ C¹([0, ∞)) is a non-negative and non-increasing function satisfying

(A2) The function is of positive type in the following sense:

for all v ∈ C¹([0, ∞)) and t > 0.

(A3) If n = 1, 2, then 1 < p, q < ∞. If n ≥ 3, then

Remark 1.1. It is clear that g(t) = εe ^-t (0 < ε < 1) satisfies the assumptions (A1) and (A2).

Based on the method of Faedo-Galerkin and Banach contraction mapping principle, the local existence and uniqueness of the problem (1.2) have been established in [8, 18, 25, 26] as follows.

Theorem 1.1. Under the assumptions (A1)-(A3), let the initial data , (u₁, v₁) ∈ L²(Ω) × L²(Ω). Then the problem (1.2) has a unique local solution

for the maximum existence time T, where T ∈ (0, ∞].

Our main blow-up result for the problem (1.2) with arbitrarily positive initial energy is stated as follows.

Theorem 1.2. Under the assumptions (A1)-(A3), if and the initial data and (u₁, v₁) ∈ L²(Ω) × L²(Ω) satisfy

(1.3)

(1.4)

(1.5)

(1.6)

then the solution of the problem (1.2) blows up in finite time T < ∞, it means

(1.7)

where χ is the constant of the Poincaré's inequality on Ω, , energy functional E(t) and I(u, v) are defined as

(1.8)

(1.9)

and .

The rest of this paper is organized as follows. In Section 2, we introduce some lemmas needed for the proof of our main results. The proof of our main results is presented in Section 3.

2 Preliminaries

In this section, we introduce some lemmas which play a crucial role in proof of our main result in next section.

Lemma 2.1. E(t) is a non-increasing function.

Proof. By differentiating (1.9) and using (1.2) and (A1), we get

(2.1)

Thus, Lemma 2.1 follows at once. At the same time, we have the following inequality:

(2.2)

Lemma 2.2. Assume that g(t) satisfies assumptions (A1) and (A2), H(t) is a twice continuously differentiable function and satisfies

(2.3)

for every t ∈ [0, T₀), and (u(x, t), v(x, t)) is the solution of the problem (1.2).

Then the function H(t) is strictly increasing on [0, T₀).

Proof. Consider the following auxiliary ODE

(2.4)

for every t ∈ [0, T₀).

It is easy to see that the solution of (2.4) is written as follows

(2.5)

for every t ∈ [0, T₀).

By a direct computation, we obtain

for every t ∈ [0, T₀).

Because g(t) satisfies (A2), then h'(t) ≥ 0, which implies that h(t) ≥ h(0) = H(0). Moreover, we see that H'(0) > h'(0).

Next, we show that

(2.6)

Assume that (2.6) is not true, let us take

By the continuity of the solutions for the ODES (2.3) and (2.4), we see that t₀ > 0 and H' (t₀) = h' (t₀), and have

which yields

This contradicts H'(t₀) = h'(t₀). Thus, we have H'(t) > h' (t) ≥ 0, which implies our desired result. The proof of Lemma 2.2 is complete.

Lemma 2.3. Suppose that , (u₁, v₁) ∈ L²(Ω) × L²(Ω) satisfies

(2.7)

If the local solution (u(t), v(t)) of the problem (1.2) exists on [0, T) and satisfies

(2.8)

then is strictly increasing on [0, T ).

Proof. Since , and (u(t), v(t)) is the local solution of problem (1.2), by a simple computation, we have

which yields

Therefore, by Lemma 2.2, the proof of Lemma 2.3 is complete.

Lemma 2.4. If , (u₁, v₁) ∈ L²(Ω) × L²(Ω) satisfy the assumptions in Theorem 1.2, then the solution (u(x, t), v(x, t)) of problem (1.2) satisfies

(2.9)

(2.10)

for every t ∈ [0, T).

Proof. We will prove the lemma by a contradiction argument. First we assume that (2.9) is not true over [0, T), it means that there exists a time t₁ such that

(2.11)

Since I (u(t, x), v(t, x)) < 0 on [0, t₁), by Lemma 2.3 we see that is strictly increasing over [0, t₁), which implies

By the continuity of on t, we have

(2.12)

On the other hand, by (2.2) we get

(2.13)

It follows from (1.9) and (2.11) that

(2.14)

Thus, by the Poincaré's inequality and , we see that

(2.15)

Obviously, (2.15) contradicts to (2.12). Thus, (2.9) holds for every t ∈ [0, T).

By Lemma 2.3, it follows that is strictly increasing on [0, T), which implies

for every t ∈ [0, T). The proof of Lemma 2.4 is complete.

3 The proof of Theorem 1.2

To prove our main result, we adopt the concavity method introduced by Levine, and define the following auxiliary function:

(3.1)

where t₂, t₃ and a are certain positive constants determined later.

Proof of Theorem 1.2. By direct computation, we obtain

(3.2)

and

(3.3)

By the Young's inequality, for any ε > 0, we have

Taking , by (1.6), (2.2), (3.3), (3.4), Lemma 2.3 and the Poincaré's in-equality, we obtain

(3.5)

which means that G"(t) > 0 for every t ∈ (0, T).

Since G'(0) ≥ 0 and G(0) ≥ 0, thus we obtain that G' (t) and G(t) are strictly increasing on [0, T).

It follows from (1.6) and that

Thus, we can choose a to satisfy

Set

By (3.2) and a simple computation, for all s ∈ R, we have

which implies that B² - AC ≤ 0.

Since we assume that the solution (u(t, x), v(t, x)) to the problem (1.2) exists for every t ∈ [0, T), then for t ∈ [0, T), one has

and

which yields

Let . As , we see that

(3.6)

for every t ∈ [0, T), which means that the function G ^-β is concave.

Let t₂ and t₃ satisfy

from which, we deduce that

Since G ^-β is a concave function and G(0) > 0, we obtain that

(3.7)

thus

(3.8)

Therefore, there exists a finite time such that

The proof of Theorem 1.2 is complete.