Global Existence and General Decay of Solutions for a Quasilinear System with Degenerate Damping Terms

Abstract

In this work, we consider a quasilinear system of viscoelastic equations with degenerate damping, dispersion, and source terms under Dirichlet boundary condition. Under some restrictions on the initial datum and standard conditions on relaxation functions, we study global existence and general decay of solutions. The results obtained here are generalization of the previous recent work.

1. Introduction

Let be a bounded domain with a sufficiently smooth boundary in We investigate a quasilinear system of two viscoelastic equations in the presence of degenerate damping, dispersion, and source terms, namely, where, , and are positive relaxation functions which will be specified later. and are the degenerate damping term and the dispersion term, respectively.

By taking in which and

It is simple to show that where

To motivate our problem (1), it can trace back to the initial boundary value problem for the single viscoelastic equation of the form

This type problem appears a variety of mathematical models in applied science. For instance, in the theory of viscoelasticity, physics, and material science, problem (5) has been studied by various authors, and several results concerning blow-up and energy decay have been studied case (). For example, Liu [1] studied a general decay of solutions case . Messaoudi and Tatar [2] applied the potential well method to indicate the global existence and uniform decay of solutions ( instead of ). Furthermore, the authors obtained a blow-up result for positive initial energy. Wu [3] studied a general decay of solution case (). Later, Wu [4] studied the same problem case and discussed the decay rate of solution energy. Recently, Yang et al. [5] proved the existence of global solution and asymptotic stability result without restrictive conditions on the relaxation function at infinity case ().

In case and without dispersion term, problem (5) has been investigated by Song [6], and the blow-up result for positive initial energy has been proved.

For a coupled system, He [7] investigated the following problem where The author proved general and optimal decay of solutions. Then, in [8], the author investigated the same problem without damping term and established a general decay of solutions. Furthermore, the author obtained a blow-up of solutions for negative initial energy. In addition, problem (1) with in case and without dispersion term, Wu [9] proved a general decay of solutions. Later, Pișkin and Ekinci [10] studied a general decay and blow-up of solutions with nonpositive initial energy for problem (1) case (Kirchhoff-type instead of and without dispersion term). In recent years, some other authors investigate the hyperbolic type system with degenerate damping term (see [11–14]).

The rest of the paper is arranged as follows: in Section 2, as preliminaries, we give necessary assumptions and lemmas that will be used later and local existence theorem without proof. In Section 3, we prove the global existence of solution. In the last section, we studied the general decay of solutions.

2. Preliminaries

We begin this section with some assumptions, notations, lemmas, and theorems. Denote the standart norm by and norm by

To state and prove our result, we need some assumptions:

(A1) Regarding is functions and satisfies and nonincreasing differentiable positive functions and such that

(A2) For the nonlinearity, we assume that

(A3) Assume that satisfies

In addition, we present some notations:

Lemma 1 (Sobolev-Poincare inequality) [15]. Let be a number with or and then there is a constant such that Now, we state the local existence theorem that can be established by combining arguments of [7, 10].

Theorem 2. Assume that (A1)-(A3) and (2) hold. Let and be given. Then, for some , problem (1) has a unique local weak solution in the following class:

We define the energy function as follows:

Also, we define

By computation, we get

3. Global Existence

In this part, in order to state and prove the global existence of solution (1), we firstly give two lemmas.

Lemma 3 [16]. Assume that (4) holds. Then, there exist such that for the solution ,

Lemma 4. Let , Suppose that (A1)-(A3) hold. If then

Proof. We have and by continuity of about , there exist a maximal time such that Let be as follows: By using (8), (9), and (A1), we get From (7) and (10), we have By (11) and (12), we infer that Thus, from (8), we obtain which contradicts to (13). Thus, on .

Theorem 5. Suppose that the conditions of Lemma 4 hold, then the solution (1) is bounded and global in time.

Proof. We have Thus, where positive constant depends only on This implies that the solution of problem (1) is global in time.

4. General Decay of Solutions

This section is devoted to show the decay of solution (1). Set where and are positive constants and

Lemma 6. For which is small enough while is large enough, the relation holds for two positive constants and

Proof. As references [1, 10], it can be show easily that and are equivalent in the sense that and are positive constants, depending on and

Lemma 7 [3]. Assume that (12) holds. Let be the solution of problem (1). Then, for we get

Lemma 8 [16]. Let (A1)-(A3) hold. Assume that , , be given and satisfying (12). Then, throughout the solution of (1), there exist two positive constants and such that for any and for all

Lemma 9. Let , , be given and satisfying (12). Suppose that (A1)-(A3) hold. Then, for each , the functional verifies, throughout the solution of (1) where

Proof. By applying (18) and Eq.(1) and getting in (18), we have For estimating the seventh term in the right side of (22) as follows (see [17]): By exploiting Young’s inequality and the assumption that , for Similarly with By estimating the following terms in (22), we have Exploiting Young’s inequality, Hölder’s inequality, Sobolev-Poincare inequality, (A3), and (15) for one has where By inserting (27) and (28) into (26), we have Similarly, for , we have Thus, inserting (24) and (25) and (29) and (30) into (22), we obtain where At this moment, choosing , and picking and small enough such that Consequently, (31) yields In order to estimate the , we set Then, by using equations (1), we have For the first term of (33), by applying (A1), Hölder’s inequality, and Young’s inequality, we deduce Then, in order to estimate the following term, we seperate such that where By Hölder’s inequality, Young’s inequality, (15), and (21), we get From (A1) assumption, Hölder’s inequality, and Young’s inequality, we get In order to estimate the forth term, we use Young’s inequality, Sobolev-Poincare inequality, Hölder’s inequality, and (A1) assumption Combining these estimates (34)-(40) and (33) becomes Similarly, let then Since the function is positive, then for any , Hence, we conclude from (17), (10), (32), (41), and (42) that where By using Lemma 8 and (15) for the last two terms of (43), we obtain At this point, we choose and which are small enough, and we have Further, we pick so small and Once is fixed, we choose that is sufficently large so that Consequently, for all , we reach at where there are positive constants .

Now, we are ready to state our stability result.

Theorem 10. Suppose that (4) and (A1)-(A3) hold, and that and satisfy and Then for each, the energy of (1) satisfies where and and are positive constants

Proof. Multiplying (46) by , we get Applying (A2) and and since by (10), we obtain That is And here, is equivalent to due to (20), and is a positive constant. A simple integration of (50) leads to This completes the proof.

5. Conclusion

As far as we know, there have not been any global existences and general decay results in the literature known for quasilinear viscoelastic equations with degenerate damping terms. Our work extends the works for some quasilinear viscoelastic equations treated in the literature to the quasilinear viscoelastic equation with degenerate damping terms.

Data Availability

No data were used to support the study.

Disclosure

Title for this paper “Global existence and general decay of solutions for a quasilinear system with degenerate damping terms” has been submitted in “Conference Proceeding of 9th International Eurasian Conference on Mathematical Sciences and Applications.”

Conflicts of Interest

The authors declare that they have no competing interests.

Acknowledgments

The authors would like to thank the handling editor and the referees for their relevant remarks and corrections in order to improve the final version.