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.
Keywords:
viscoelastic equations; blow up; positive initial energy1 Introduction
In this work, we study the following wave equations with nonlinear viscoelastic term
where Ω is a bounded domain of R^{n }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 nonexistence of a global solution, Levine [11] firstly showed that the solutions with negative initial energy are nonglobal for some abstract wave equation with linear damping. Later Levine and Serrin [12] studied blowup 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 blowup 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 blowup result with negative energy. Furthermore, he improved his blowup 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 blowup 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
where
We next state some assumptions on g(s) and real numbers p > 1, q > 1.
(A1) g ∈ C^{1}([0, ∞)) is a nonnegative and nonincreasing function satisfying
(A2) The function is of positive type in the following sense:
for all v ∈ C^{1}([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 FaedoGalerkin 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 , (u_{1}, v_{1}) ∈ L^{2}(Ω) × L^{2}(Ω). Then the problem (1.2) has a unique local solution
for the maximum existence time T, where T ∈ (0, ∞].
Our main blowup 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 (u_{1}, v_{1}) ∈ L^{2}(Ω) × L^{2}(Ω) satisfy
then the solution of the problem (1.2) blows up in finite time T < ∞, it means
where χ is the constant of the Poincaré's inequality on Ω, , energy functional E(t) and I(u, v) are defined as
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 nonincreasing function.
Proof. By differentiating (1.9) and using (1.2) and (A1), we get
Thus, Lemma 2.1 follows at once. At the same time, we have the following inequality:
Lemma 2.2. Assume that g(t) satisfies assumptions (A1) and (A2), H(t) is a twice continuously differentiable function and satisfies
for every t ∈ [0, T_{0}), 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, T_{0}).
Proof. Consider the following auxiliary ODE
for every t ∈ [0, T_{0}).
It is easy to see that the solution of (2.4) is written as follows
for every t ∈ [0, T_{0}).
By a direct computation, we obtain
for every t ∈ [0, T_{0}).
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
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 t_{0 }> 0 and H' (t_{0}) = h' (t_{0}), and have
which yields
This contradicts H'(t_{0}) = h'(t_{0}). 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 , (u_{1}, v_{1}) ∈ L^{2}(Ω) × L^{2}(Ω) satisfies
If the local solution (u(t), v(t)) of the problem (1.2) exists on [0, T) and satisfies
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 , (u_{1}, v_{1}) ∈ L^{2}(Ω) × L^{2}(Ω) satisfy the assumptions in Theorem 1.2, then the solution (u(x, t), v(x, t)) of problem (1.2) satisfies
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 t_{1 }such that
Since I (u(t, x), v(t, x)) < 0 on [0, t_{1}), by Lemma 2.3 we see that is strictly increasing over [0, t_{1}), which implies
By the continuity of on t, we have
On the other hand, by (2.2) we get
It follows from (1.9) and (2.11) that
Thus, by the Poincaré's inequality and , we see that
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:
where t_{2}, t_{3 }and a are certain positive constants determined later.
Proof of Theorem 1.2. By direct computation, we obtain
and
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 inequality, we obtain
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 B^{2 } 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
for every t ∈ [0, T), which means that the function G ^{β }is concave.
Let t_{2 }and t_{3 }satisfy
from which, we deduce that
Since G ^{β }is a concave function and G(0) > 0, we obtain that
thus
Therefore, there exists a finite time such that
The proof of Theorem 1.2 is complete.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
MJ and CL carried out all studies in the paper. ZR participated in the design of the study in the paper.
Acknowledgements
This work is supported in part by NSF of PR China (11071266) and in part by Natural Science Foundation Project of CQ CSTC (2010BB9218).
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