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

        Abstract

        In this paper, we consider the following viscoelastic equations

        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equa_HTML.gif

        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 energy

        1 Introduction

        In this work, we study the following wave equations with nonlinear viscoelastic term
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ1_HTML.gif
        (1.1)

        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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equb_HTML.gif
        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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equc_HTML.gif

        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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ2_HTML.gif
        (1.2)
        where
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equd_HTML.gif

        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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Eque_HTML.gif
        (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:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equf_HTML.gif

        for all vC1([0, ∞)) and t > 0.

        (A3) If n = 1, 2, then 1 < p, q < ∞. If n ≥ 3, then
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equg_HTML.gif
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equh_HTML.gif

        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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equi_HTML.gif

        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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ3_HTML.gif
        (1.3)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ4_HTML.gif
        (1.4)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ5_HTML.gif
        (1.5)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ6_HTML.gif
        (1.6)
        then the solution of the problem (1.2) blows up in finite time T < ∞, it means
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ7_HTML.gif
        (1.7)
        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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ8_HTML.gif
        (1.8)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ9_HTML.gif
        (1.9)

        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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ10_HTML.gif
        (2.1)
        Thus, Lemma 2.1 follows at once. At the same time, we have the following inequality:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ11_HTML.gif
        (2.2)
        Lemma 2.2. Assume that g(t) satisfies assumptions (A1) and (A2), H(t) is a twice continuously differentiable function and satisfies
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ12_HTML.gif
        (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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ13_HTML.gif
        (2.4)

        for every t ∈ [0, T0).

        It is easy to see that the solution of (2.4) is written as follows
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ14_HTML.gif
        (2.5)

        for every t ∈ [0, T0).

        By a direct computation, we obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equj_HTML.gif

        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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ15_HTML.gif
        (2.6)
        Assume that (2.6) is not true, let us take
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equk_HTML.gif
        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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equl_HTML.gif
        which yields
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equm_HTML.gif

        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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ16_HTML.gif
        (2.7)
        If the local solution (u(t), v(t)) of the problem (1.2) exists on [0, T) and satisfies
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ17_HTML.gif
        (2.8)

        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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equn_HTML.gif
        which yields
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equo_HTML.gif

        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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ18_HTML.gif
        (2.9)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ19_HTML.gif
        (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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ20_HTML.gif
        (2.11)
        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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equp_HTML.gif
        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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ21_HTML.gif
        (2.12)
        On the other hand, by (2.2) we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ22_HTML.gif
        (2.13)
        It follows from (1.9) and (2.11) that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ23_HTML.gif
        (2.14)
        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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ24_HTML.gif
        (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 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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equq_HTML.gif

        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:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ25_HTML.gif
        (3.1)

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

        Proof of Theorem 1.2. By direct computation, we obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ26_HTML.gif
        (3.2)
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ27_HTML.gif
        (3.3)
        By the Young's inequality, for any ε > 0, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equr_HTML.gif
        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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ28_HTML.gif
        (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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_IEq9_HTML.gif that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equs_HTML.gif
        Thus, we can choose a to satisfy
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equt_HTML.gif
        Set
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equu_HTML.gif
        By (3.2) and a simple computation, for all sR, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equv_HTML.gif

        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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equw_HTML.gif
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equx_HTML.gif
        which yields
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equy_HTML.gif
        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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ29_HTML.gif
        (3.6)

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

        Let t2 and t3 satisfy
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equz_HTML.gif
        from which, we deduce that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equaa_HTML.gif
        Since G is a concave function and G(0) > 0, we obtain that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ30_HTML.gif
        (3.7)
        thus
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equ31_HTML.gif
        (3.8)
        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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-6/MediaObjects/13661_2011_Article_6_Equab_HTML.gif

        The proof of Theorem 1.2 is complete.

        Declarations

        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).

        Authors’ Affiliations

        (1)
        College of Mathematics and Statistics, Chongqing University

        References

        1. Fabrizio M, Morro A: Mathematical problems in linear viscoelasticity. SIAM Studies in Applied Mathematics Philadelphia 1992., 12:
        2. Cavalcanti MM, Cavalcanti VND, Lasiecka I: Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping source interaction. J Diff Equ 2007, 236(2):407-459. 10.1016/j.jde.2007.02.004View ArticleMathSciNetMATH
        3. Cavalcanti MM, Cavalcanti VND, Martinez P: Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term. J Diff Equ 2004, 203(1):119-158. 10.1016/j.jde.2004.04.011View ArticleMathSciNetMATH
        4. Vitillaro E: A potential well theory for the wave equation with nonlinear source and boundary damping terms. Glasg Math J 2002, 44(3):375-395. 10.1017/S0017089502030045View ArticleMathSciNetMATH
        5. Vitillaro E: Global existence for the wave equation with nonlinear boundary damping and source terms. J Diff Equ 2002, 186(1):259-298. 10.1016/S0022-0396(02)00023-2View ArticleMathSciNet
        6. Aassila M, Cavalcanti MM, Cavalcanti VND: Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term. Calc Var Partial Diff Equ 2002, 15(2):155-180. 10.1007/s005260100096View ArticleMathSciNetMATH
        7. Aassila M, Cavalcanti MM, Soriano JA: Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star-shaped domain. SIAM J Control Optim 2000, 38(5):1581-1602. 10.1137/S0363012998344981View ArticleMathSciNetMATH
        8. Cavalcanti MM, Cavalcanti VND, Soriano JA: Existence and uniform decay rate for viscoelastic problems with nonlinear boundary damping. Diff Integ Equ 2001, 14: 85-116.MathSciNetMATH
        9. Alves CO, Cavalcanti MM: On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source. Calc Var Partial Diff Equ 2009, 34(3):377-411. 10.1007/s00526-008-0188-zView ArticleMathSciNetMATH
        10. Cavalcanti MM, Cavalcanti VND: Existence and asymptotic stability for evolution problems on manifolds with damping and source terms. J Math Anal Appl 2004, 291(1):109-127. 10.1016/j.jmaa.2003.10.020View ArticleMathSciNetMATH
        11. Levine HA: Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM J Math Anal 1974, 5: 138-146. 10.1137/0505015View ArticleMathSciNetMATH
        12. Levine HA, Serrin J: Global nonexistence theorems for quasilinear evolution equation with dissipation. Arch Ration Mech Anal 1997, 137: 341-361. 10.1007/s002050050032View ArticleMathSciNetMATH
        13. Pucci P, Serrin J: Global nonexistence for abstract evolution equation with positive initial energy. J Diff Equ 1998, 150: 203-214. 10.1006/jdeq.1998.3477View ArticleMathSciNetMATH
        14. Levine HA, Todorova G: Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy. Proc Am Math Soc 2001, 129: 793-805. 10.1090/S0002-9939-00-05743-9View ArticleMathSciNetMATH
        15. Todorova G, Vitillaro E: Blow-up for nonlinear dissipative wave equations in R n . J Math Anal Appl 2005, 303: 242-257. 10.1016/j.jmaa.2004.08.039View ArticleMathSciNetMATH
        16. Gazzola F, Squassina M: Global solutions and finite time blow up for damed semilinear wave equations. Ann Inst H Poincare Anal NonLineaire 2006, 23(2):185-207. 10.1016/j.anihpc.2005.02.007View ArticleMathSciNetMATH
        17. Zeng R, Mu CL, Zhou SM: A blow up result for Kirchhoff type equations with high energy. Math Methods Appl Sci 2011, 34(4):479-486.MathSciNetMATH
        18. Cavalcanti MM, Cavalcanti VND, Soriano JA: Exponential decay for the solution of the semilinear viscoelastic wave equations with localized damping. Electron J Diff Equ 2002, 44: 1-14.MathSciNet
        19. Cavalcanti MM, Oquendo HP: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J Control Optim 2003, 42(4):1310-1324. 10.1137/S0363012902408010View ArticleMathSciNetMATH
        20. Messaoudi SA: Blow up and global existence in a nonlinear viscoelastic wave equation. Math Nachr 2003, 260: 58-66. 10.1002/mana.200310104View ArticleMathSciNetMATH
        21. Messaoudi SA: Blow up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation. J Math Anal Appl 2006, 320: 902-915. 10.1016/j.jmaa.2005.07.022View ArticleMathSciNetMATH
        22. Wang YJ, Wang YF: Exponential energy decay of solutions of viscoelastic wave equations. J Math Anal Appl 2008, 347: 18-25. 10.1016/j.jmaa.2008.05.098View ArticleMathSciNetMATH
        23. Wang YJ: A global nonexistence theorem for viscoelastic equations with arbitrary positive initial energy. Appl Math Lett 2009, 22: 1394-1400. 10.1016/j.aml.2009.01.052View ArticleMathSciNetMATH
        24. Levine HA: Instability and nonexistence of global solutions of nonlinear wave equation of the form Pu tt = Δ u + F ( u ). Trans Am Math Soc 1974, 192: 1-21.MATH
        25. Cavalcanti MM, Cavalcanti VND, Ferreira J: Existence and uniform decay for nonlinear viscoelastic equation with strong damping. Math Meth Appl Sci 2001, 24: 1043-1053. 10.1002/mma.250View ArticleMathSciNetMATH
        26. Georgiev V, Todorova G: Existence of a solution of the wave equation with nonlinear damping and source term. J Diff Equ 1994, 109: 295-308. 10.1006/jdeq.1994.1051View ArticleMathSciNetMATH

        Copyright

        © Ma et al; licensee Springer. 2011

        This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.