Abstract
We consider the initial boundary value problem of a nonlinear viscoelastic equation of Kirchhoff-type with nonlinear damping and velocity-dependent material density. We establish a nonexistence result of global solutions with positive initial energy and negative initial energy, respectively.
1. Introduction
In this paper we study the global nonexistence for solutions of the nonlinear viscoelastic equation problem with nonlinear damping and velocity-dependent material density as follows:where is a bounded domain in with smooth boundary so that the divergence theorem can be applied and is a positive function that represents the kernel of the memory term. Here , and are positive constants. In this paper, we take , and , where , and are positive constants.
This model describes a small amplitude vibration of an elastic string (see Kirchhoff [1]) and derives from the description of the vibrations of thin rods whose density depends on the velocity (see, e.g., [2]). The motivation of our work is the results regarding viscoelastic equations of Kirchhoff type by Zhang et al. [3]. They study the global existence and general decay of the energy for solutions to a nonlinear viscoelastic equation with nonlinear localized damping and velocity-dependent material densityThey also give a detailed review of literatures. However, they do not take the global nonexistence of the solution to (4) with nonlinear source into consideration. For more information about this, readers can refer to [3–18] and references therein. It is well known that the nonlinear source term causes global nonexistence of solutions when either the condition or holds in (1)(see [5–15] and references cited therein). Liu and Wang [19] consider the initial boundary value problem of and they establish a blow up result for certain solutions with nonpositive initial energy as well as positive initial energy. Said-Houari [20] studies the following system of nonlinear viscoelastic wave equations:They prove that the energy of the system will grow up as an exponential function as time goes to infinity, provided that the initial data are large enough. They do not get the blow-up result. Tahamtani and Pwyravi [6] study the following system:and they prove uniform decay of solution energy under some restrictions on the initial data and the relaxation functions. Moreover, they establish a growth result for certain solutions with positive initial energy, but they do not get the blow-up result. Therefore, a problem of whether the solution of problem (1)-(3) still blows up in finite time for appropriately bounded initial energy when introducing both presences of nonlinear weak damping term and linear strong damping term (i.e., ) arises. For the following special case, Gazzola and Squassina [21] consider the initial boundary problem:and obtain the polynomial decay and finite time blow-up result under certain initial values. Later, Gerbi and Houari [22] obtain the exponential decay based on a small perturbation of energy. Chen and Liu [23] extend the linear damping term to the nonlinear damping term and obtain the energy decay rate and the exponential growth. Similarly, [24] is also under the dynamic boundary conditions. From the physics points of view, the strong damping term and the nonlinear dissipative damping term play a dissipative or inhibitive part in the energy accumulation in the configurations. But the nonlinear source term leads to energy gathering in the configurations. If the energy accumulation arising from the nonlinear source term and other nonlinear factors cannot be dissipated synchronously, the remaining energy accumulation may cause the configurations to break or burn out in finite time; i.e., the solutions of problem (1)-(3) blow up in finite time. However introducing both presences of nonlinear weak damping term and linear strong damping term makes the problem interesting but difficult. Indeed, a strong impact of dissipative terms could make the existence of global solutions easier since they play the role of stabilizing terms and their smoothing effect makes the blow-up more difficult [11, 17, 25]. The most frequently used technique "concavity argument" in the proof of blow-up is no longer effective, and the technique in papers mentioned above cannot be used directly here either. At present only a few results are known for the interaction between the weak damping term and the strong damping term .
The main purpose of this paper is to investigate the nonexistence result of global solutions for problem (1)-(3) with both terms and . More precisely, we shall show global nonexistence results for problem (1)-(3), and we should overcome the difficulties brought by the treatment of the nonlinear terms and interaction among the damping term , memory term , and the source term . The outline of this article is as follows. In Section 2, we introduce some notations, assumptions, and preliminaries. In Section 3, we show the main results of this article. For simplicity, we assume .
2. Preliminaries
In this section, we give some assumptions and known results in order to state the main results of this article. Throughout this article, the following notations are used for precise statements: denotes the usual space of all -functions on with norm and inner product . For simplicity, we denote and we take . The constant used throughout this paper is positive generic constants, which may vary in different situation. We also denote that
First, we present the following assumptions:
(A1) , , for all .
(A2) is a function satisfyingand here and are positive constants.
(A3) if , and if .
(A4) and if , and if .
Next, we present the following local existence theorem that can be found in [3](see also [26]).
Theorem 1 (see [3]). Supposing that (A1)-(A4) hold and that , and , then problem (1)-(3) admits a unique solution
Now, we define the energy of the problem (1)-(3) bywhereSimilar to the proof of Lemma 4.1 in [12] and the proof of Lemma 3.1 in [20], multiplying the first equation of (1) by , integrating over , and using integrating by parts, we have the following results.
Lemma 2. is a nonincreasing function on andFrom (A2) and Poincare inequality, we getfor , where , is the Poincare constant, andIt is easy to verify that has a maximum at and the maximum value is . Before we prove the main result, we need the following lemma, which is similar to the proof of Lemma 5.1 in [12], the proof of Lemma 3.2 in [15], and the proof of Lemma 3.2 in [20].
Lemma 3. Assume that (A1)-(A4) hold, , and let be a solution of problem (1)-(3) with initial data satisfying and . Then there exists a constant , such that
3. Main Results
In this section, we prove blow-up result for the solution of problem (1)-(3) with positive but appropriately bounded and negative initial energy, respectively. In order to state our main result, we make an extra assumption on :
Theorem 4. Let . Assuming that (A1)-(A4) and (18) hold, , and , then any solution of problem (1)-(3) with initial data satisfying and will blow up in finite time.
Proof. We set where . From (14) and (19), we getand then is an increasing function andOn the other hand, by Lemma 3, we haveHence, combining (21) and (22) with the embedding , we haveWe setand then definewhere are small parameters to be chosen later. By the definition of the solution, we haveUsing Schwarz’s inequality and Young’s inequality, (26) takes on the formAdding the term and using the definition of , then (27) becomesWe denote ,, and . By and (18), we observe that , and . Due to the restriction on , we have , and then (28) becomesUsing the fact that by Lemma 3, we getwhere and . Moreover, by the assumption of , Lemma 3, , the definition of , and (18), we see that andThus, (29) yields By Hölder inequality and (23) we haveConsidering (23), Young’s inequality, and the fact that , we getwhere . Now, we make and satisfyand then we haveFurthermore, from (34) and (23), we haveBy differentiating (25), from (32) and (37), we getLetting , decomposing in (38) by , and noting from (23) and (38), we find thatChoosing small enough so that and , from (40), we haveTherefore, is a nondecreasing function for . Letting in (25) small enough, we get . Consequently, we obtain for .
We claim the inequalityFor the proof of (42), we consider two alternatives:
(i) If there exists a so that , thenThus (42) follows (43).
(ii) If there exists a so that , since by (36), then we deduce from (25), Young inequality, Hölder inequality, and the embedding thatfor . We take , by the restrictions on in (35) and (36), then andThus from (44), (46), and (23), we haveThis inequality together with (41) implies (42).
Then, by integrating both sides of (42) over , it follows that there exists a such thatThis limit combining with (47), (43), and (23) gives This theorem is proved.
Theorem 5. Let . Assuming that (A1)-(A3), , and hold and , then the local solution of problem (1)-(3) blows up in finite time.
Proof. Setting instead of in (19) and then applying the same arguments as that in Theorem 4, we get the desired result.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (No. 11801145).