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Exponential Decay of Energy for Some Nonlinear Hyperbolic Equations with Strong Dissipation


The initial boundary value problem for a class of hyperbolic equations with strong dissipative term in a bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set in and showing the exponential decay of the energy of global solutions through the use of an important lemma of V. Komornik.

1. Introduction

We are concerned with the global solvability and exponential asymptotic stability for the following hyperbolic equation in a bounded domain:


with initial conditions


and boundary condition


where is a bounded domain in with a smooth boundary , and are real numbers, and is a divergence operator (degenerate Laplace operator) with , which is called a -Laplace operator.

Equations of type (1.1) are used to describe longitudinal motion in viscoelasticity mechanics and can also be seen as field equations governing the longitudinal motion of a viscoelastic configuration obeying the nonlinear Voight model [14].

For , it is well known that the damping term assures global existence and decay of the solution energy for arbitrary initial data [46]. For , the source term causes finite time blow up of solutions with negative initial energy if [7].

In [810], Yang studied the problem (1.1)–(1.3) and obtained global existence results under the growth assumptions on the nonlinear terms and initial data. These global existence results have been improved by Liu and Zhao [11] by using a new method. As for the nonexistence of global solutions, Yang [12] obtained the blow up properties for the problem (1.1)–(1.3) with the following restriction on the initial energy , where and , and are some positive constants.

Because the -Laplace operator is nonlinear operator, the reasoning of proof and computation are greatly different from the Laplace operator . By means of the Galerkin method and compactness criteria and a difference inequality introduced by Nakao [13], Ye [14, 15] has proved the existence and decay estimate of global solutions for the problem (1.1)–(1.3) with inhomogeneous term and .

In this paper we are going to investigate the global existence for the problem (1.1)–(1.3) by applying the potential well theory introduced by Sattinger [16], and we show the exponential asymptotic behavior of global solutions through the use of the lemma of Komornik [17].

We adopt the usual notation and convention. Let denote the Sobolev space with the norm and denote the closure in of . For simplicity of notation, hereafter we denote by the Lebesgue space norm, denotes norm, and write equivalent norm instead of norm . Moreover, denotes various positive constants depending on the known constants, and it may be different at each appearance.

2. The Global Existence and Nonexistence

In order to state and study our main results, we first define the following functionals:


for . Then we define the stable set by




We denote the total energy associated with (1.1)–(1.3) by


for , , and is the total energy of the initial data.

Definition 2.1.

The solution is called the weak solution of the problem (1.1)–(1.3) on , if satisfy


and in , in .

We need the following local existence result, which is known as a standard one (see [14, 18, 19]).

Theorem 2.2.

Suppose that if and if . If , then there exists such that the problem (1.1)–(1.3) has a unique local solution in the class


For latter applications, we list up some lemmas.

Lemma 2.3 (see [20, 21]).

Let , then , and the inequality holds with a constant depending on , and , provided that, if and .

Lemma 2.4.

Let be a solution to problem (1.1)–(1.3). Then is a nonincreasing function for and



By multiplying (1.1) by and integrating over , we get


which implies from (2.4) that


Therefore, is a nonincreasing function on .

Lemma 2.5.

Let ; if the hypotheses in Theorem 2.2 hold, then .




so, we get


Let , which implies that


As , an elementary calculation shows that


Hence, we have from Lemma 2.3 that


We get from the definition of that

Lemma 2.6.

Let , then



By the definition of and , we have the following identity:


Since , so we have . Therefore, we obtain from (2.16) that


In order to prove the existence of global solutions for the problem (1.1)-(1.3), we need the following lemma.

Lemma 2.7.

Suppose that if and if . If , and , then , for each .


Assume that there exists a number such that on and . Then, in virtue of the continuity of , we see that . From the definition of and the continuity of and in , we have either




It follows from (2.4) that


So, case (2.18) is impossible.

Assume that (2.19) holds, then we get that


We obtain from that .



consequently, we get from (2.20) that


which contradicts the definition of . Therefore, case (2.19) is impossible as well. Thus, we conclude that on .

Theorem 2.8.

Assume that if and if . is a local solution of problem (1.1)–(1.3) on . If , and , then the solution is a global solution of the problem (1.1)–(1.3).


It suffices to show that is bounded independently of .

Under the hypotheses in Theorem 2.8, we get from Lemma 2.7 that on . So formula (2.15) in Lemma 2.6 holds on . Therefore, we have from (2.15) and Lemma 2.4 that


Hence, we get


The above inequality and the continuation principle lead to the global existence of the solution, that is, . Thus, the solution is a global solution of the problem (1.1)–(1.3).

Now we employ the analysis method to discuss the blow-up solutions of the problem (1.1)–(1.3) in finite time. Our result reads as follows.

Theorem 2.9.

Suppose that if and if . If , assume that the initial value is such that




with is a positive Sobolev constant. Then the solution of the problem (1.1)–(1.3) does not exist globally in time.


On the contrary, under the conditions in Theorem 2.9, let be a global solution of the problem (1.1)–(1.3); then by Lemma 2.3, it is well known that there exists a constant depending only on , and such that for all .

From the above inequality, we conclude that


By using (2.28), it follows from the definition of that




we denote the right side of (2.29) by , then


We have


Letting , we obtain .

As , we have


Consequently, the function has a single maximum value at , where


Since the initial data is such that satisfies


Therefore, from Lemma 2.4 we get


At the same time, by (2.29) and (2.31), it is clear that there can be no time for which


Hence we have also for all from the continuity of and .

According to the above contradiction, we know that the global solution of the problem (1.1)–(1.3) does not exist, that is, the solution blows up in some finite time.

This completes the proof of Theorem 2.9.

3. The Exponential Asymptotic Behavior

Lemma 3.1 (see [17]).

Let be a nonincreasing function, and assume that there is a constant such that


then , for all .

The following theorem shows the exponential asymptotic behavior of global solutions of problem (1.1)–(1.3).

Theorem 3.2.

If the hypotheses in Theorem 2.8 are valid, then the global solutions of problem (1.1)–(1.3) have the following exponential asymptotic behavior:



Multiplying by on both sides of (1.1) and integrating over gives


where .



so, substituting the formula (3.4) into the right-hand side of (3.3) gives


By exploiting Lemma 2.3 and (2.24), we easily arrive at


We obtain from (3.6) and (2.24) that


It follows from (3.7) and (3.5) that


We have from Hölder inequality, Lemma 2.3 and (2.24) that


Substituting the estimates of (3.9) into (3.8), we conclude that


We get from Lemma 2.3 and Lemma 2.4 that


From Young inequality, Lemmas 2.3 and 2.4, and (2.24), it follows that


Choosing small enough, such that


and, substituting (3.11) and (3.12) into (3.10), we get


We let in (3.14) to get


Therefore, we have from (3.15) and Lemma 3.1 that


We conclude from , (2.4) and (3.16) that


The proof of Theorem 3.2 is thus finished.


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This paper was supported by the Natural Science Foundation of Zhejiang Province (no. Y6100016), the Science and Research Project of Zhejiang Province Education Commission (no. Y200803804 and Y200907298). The Research Foundation of Zhejiang University of Science and Technology (no. 200803), and the Middle-aged and Young Leader in Zhejiang University of Science and Technology (2008–2010).

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Correspondence to Yaojun Ye.

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Ye, Y. Exponential Decay of Energy for Some Nonlinear Hyperbolic Equations with Strong Dissipation. Adv Differ Equ 2010, 357404 (2010).

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  • Global Solution
  • Finite Time
  • Global Existence
  • Hyperbolic Equation
  • Longitudinal Motion