Exponential Decay of Energy for Some Nonlinear Hyperbolic Equations with Strong Dissipation
© Yaojun Ye. 2010
Received: 14 December 2009
Accepted: 4 August 2010
Published: 15 August 2010
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.
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 .
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  by using a new method. As for the nonexistence of global solutions, Yang  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 , 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 , and we show the exponential asymptotic behavior of global solutions through the use of the lemma of Komornik .
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
for , , and is the total energy of the initial data.
and in , in .
For latter applications, we list up some lemmas.
Let , then , and the inequality holds with a constant depending on , and , provided that, if and .
Therefore, is a nonincreasing function on .
Let ; if the hypotheses in Theorem 2.2 hold, then .
We get from the definition of that
In order to prove the existence of global solutions for the problem (1.1)-(1.3), we need the following lemma.
Suppose that if and if . If , and , then , for each .
So, case (2.18) is impossible.
We obtain from that .
which contradicts the definition of . Therefore, case (2.19) is impossible as well. Thus, we conclude that on .
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 .
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.
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 .
Letting , we obtain .
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 ).
then , for all .
The following theorem shows the exponential asymptotic behavior of global solutions of problem (1.1)–(1.3).
The proof of Theorem 3.2 is thus finished.
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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