Journal of Function Spaces
Volume 2019 (2019), Article ID 8306790, 7 pages
https://doi.org/10.1155/2019/8306790

Research Article

Global Nonexistence for a Nonlinear Viscoelastic Equation with Nonlinear Damping and Velocity-Dependent Material Density

Jian Dang, Qingying Hu, and Hongwei Zhang

Department of Mathematics, Henan University of Technology, Zhengzhou 450001, China

Correspondence should be addressed to Hongwei Zhang; whz661@163.com

Received 28 November 2018; Revised 8 February 2019; Accepted 13 February 2019; Published 26 March 2019

Academic Editor: Maria Alessandra Ragusa

Copyright © 2019 Jian Dang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 density

They 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 satisfying

and 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) by

where

Similar 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 and

From (A2) and Poincare inequality, we get

for , where , is the Poincare constant, and

It 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 get

and then is an increasing function and

On the other hand, by Lemma 3, we have

Hence, combining (21) and (22) with the embedding , we have

We set

and then define

where are small parameters to be chosen later. By the definition of the solution, we have

Using Schwarz’s inequality and Young’s inequality, (26) takes on the form

Adding the term and using the definition of , then (27) becomes

We denote ,, and . By and (18), we observe that , and . Due to the restriction on , we have , and then (28) becomes

Using the fact that by Lemma 3, we get

where and . Moreover, by the assumption of , Lemma 3, , the definition of , and (18), we see that and

Thus, (29) yields

By Hölder inequality and (23) we have

Considering (23), Young’s inequality, and the fact that , we get

where . Now, we make and satisfy

and then we have

Furthermore, from (34) and (23), we have

By differentiating (25), from (32) and (37), we get

Letting , decomposing in (38) by , and noting

from (23) and (38), we find that

Choosing small enough so that and , from (40), we have

Therefore, is a nondecreasing function for . Letting in (25) small enough, we get . Consequently, we obtain for .
We claim the inequality

For the proof of (42), we consider two alternatives:
(i) If there exists a so that , then

Thus (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 that

for . We take , by the restrictions on in (35) and (36), then

and

Thus from (44), (46), and (23), we have

This inequality together with (41) implies (42).
Then, by integrating both sides of (42) over , it follows that there exists a such that

This 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).

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