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

Abstract

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:

(1.1)

with initial conditions

(1.2)

and boundary condition

(1.3)

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 [1–4].

For , it is well known that the damping term assures global existence and decay of the solution energy for arbitrary initial data [4–6]. For , the source term causes finite time blow up of solutions with negative initial energy if [7].

In [8–10], 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:

(2.1)

for . Then we define the stable set by

(2.2)

where

(2.3)

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

(2.4)

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

(2.5)

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

(2.6)

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

(2.7)

Proof.

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

(2.8)

which implies from (2.4) that

(2.9)

Therefore, is a nonincreasing function on .

Lemma 2.5.

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

Proof.

Since

(2.10)

so, we get

(2.11)

Let , which implies that

(2.12)

As , an elementary calculation shows that

(2.13)

Hence, we have from Lemma 2.3 that

(2.14)

We get from the definition of that

Lemma 2.6.

Let , then

(2.15)

Proof.

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

(2.16)

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

(2.17)

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 .

Proof.

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

(2.18)

or

(2.19)

It follows from (2.4) that

(2.20)

So, case (2.18) is impossible.

Assume that (2.19) holds, then we get that

(2.21)

We obtain from that .

Since

(2.22)

consequently, we get from (2.20) that

(2.23)

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).

Proof.

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

(2.24)

Hence, we get

(2.25)

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

(2.26)

where

(2.27)

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

Proof.

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

(2.28)

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

(2.29)

Setting

(2.30)

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

(2.31)

We have

(2.32)

Letting , we obtain .

As , we have

(2.33)

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

(2.34)

Since the initial data is such that satisfies

(2.35)

Therefore, from Lemma 2.4 we get

(2.36)

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

(2.37)

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

(3.1)

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:

(3.2)

Proof.

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

(3.3)

where .

Since

(3.4)

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

(3.5)

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

(3.6)

We obtain from (3.6) and (2.24) that

(3.7)

It follows from (3.7) and (3.5) that

(3.8)

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

(3.9)

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

(3.10)

We get from Lemma 2.3 and Lemma 2.4 that

(3.11)

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

(3.12)

Choosing small enough, such that

(3.13)

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

(3.14)

We let in (3.14) to get

(3.15)

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

(3.16)

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

(3.17)

The proof of Theorem 3.2 is thus finished.