A blow up result for viscoelastic equations with arbitrary positive initial energy
© Ma et al; licensee Springer. 2011
Received: 5 March 2011
Accepted: 12 July 2011
Published: 12 July 2011
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.
Keywordsviscoelastic equations blow up positive initial energy
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 ).
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. . 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 , Cavalcanti and Domingos Cavalcanti . As regards non-existence of a global solution, Levine  firstly showed that the solutions with negative initial energy are non-global for some abstract wave equation with linear damping. Later Levine and Serrin  studied blow-up of a class of more generalized abstract wave equations. Then Pucci and Serrin  claimed that the solution blows up in finite time with positive initial energy which is appropriately bounded. In  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  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  established the condition for initial data with arbitrary positive initial energy such that the corresponding solution blows up in finite time. Zeng et al.  studied blowup of solutions for the Kirchhoff type equation with arbitrary positive initial energy.
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  Wang established the blow-up result for the above problem when the initial energy is high.
We next state some assumptions on g(s) and real numbers p > 1, q > 1.
for all v ∈ C1([0, ∞)) and t > 0.
Remark 1.1. It is clear that g(t) = εe -t (0 < ε < 1) satisfies the assumptions (A1) and (A2).
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.
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.
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.
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).
for every t ∈ [0, T0).
for every t ∈ [0, T0).
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).
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.
then is strictly increasing on [0, T ).
Therefore, by Lemma 2.2, the proof of Lemma 2.3 is complete.
for every t ∈ [0, T).
Obviously, (2.15) contradicts to (2.12). Thus, (2.9) holds for every t ∈ [0, T).
for every t ∈ [0, T). The proof of Lemma 2.4 is complete.
3 The proof of Theorem 1.2
where t2, t3 and a are certain positive constants determined later.
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).
which implies that B2 - AC ≤ 0.
for every t ∈ [0, T), which means that the function G -β is concave.
The proof of Theorem 1.2 is complete.
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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