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

  • Jie Ma1Email author,

    Affiliated with

    • Chunlai Mu1 and

      Affiliated with

      • Rong Zeng1

        Affiliated with

        Boundary Value Problems20112011:6

        DOI: 10.1186/1687-2770-2011-6

        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.


        viscoelastic equations blow up positive initial energy

        1 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_IEq1_HTML.gif , 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

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

        (A1) gC1([0, ∞)) is a non-negative and non-increasing function satisfying
        (A2) The function http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_IEq2_HTML.gif is of positive type in the following sense:

        for all vC1([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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_IEq3_HTML.gif , (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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_IEq4_HTML.gif and the initial data http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_IEq3_HTML.gif and (u1, v1) ∈ L2(Ω) × L2(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, http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_IEq5_HTML.gif , energy functional E(t) and I(u, v) are defined as

        and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_IEq6_HTML.gif .

        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
        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, 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

        for every t ∈ [0, T0).

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

        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
        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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_IEq3_HTML.gif , (u1, v1) ∈ L2(Ω) × L2(Ω) satisfies
        If the local solution (u(t), v(t)) of the problem (1.2) exists on [0, T) and satisfies

        then http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_IEq7_HTML.gif is strictly increasing on [0, T ).

        Proof. Since http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_IEq8_HTML.gif , 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_IEq3_HTML.gif , (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

        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
        Since I (u(t, x), v(t, x)) < 0 on [0, t1), by Lemma 2.3 we see that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_IEq7_HTML.gif is strictly increasing over [0, t1), which implies
        By the continuity of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_IEq7_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_IEq9_HTML.gif , 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_IEq7_HTML.gif 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 t2, t3 and a are certain positive constants determined later.

        Proof of Theorem 1.2. By direct computation, we obtain
        By the Young's inequality, for any ε > 0, we have
        Taking http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_IEq10_HTML.gif , by (1.6), (2.2), (3.3), (3.4), Lemma 2.3 and the Poincaré's in-equality, 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_IEq9_HTML.gif that
        Thus, we can choose a to satisfy
        By (3.2) and a simple computation, for all sR, 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
        which yields
        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_IEq11_HTML.gif . As http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_IEq12_HTML.gif , we see that

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

        Let t2 and t3 satisfy
        from which, we deduce that
        Since G is a concave function and G(0) > 0, we obtain that
        Therefore, there exists a finite time http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_IEq13_HTML.gif such that

        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).

        Authors’ Affiliations

        College of Mathematics and Statistics, Chongqing University


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