- Research Article
- Open Access
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.
- Global Solution
- Finite Time
- Global Existence
- Hyperbolic Equation
- Longitudinal Motion
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.
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.
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).
- Andrews G:On the existence of solutions to the equation . Journal of Differential Equations 1980,35(2):200-231. 10.1016/0022-0396(80)90040-6MathSciNetView ArticleGoogle Scholar
- Andrews G, Ball JM: Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity. Journal of Differential Equations 1982,44(2):306-341. 10.1016/0022-0396(82)90019-5MathSciNetView ArticleMATHGoogle Scholar
- Ang DD, Dinh PN: Strong solutions of quasilinear wave equation with non-linear damping. SIAM Journal on Mathematical Analysis 1985, 19: 337-347. 10.1137/0519024View ArticleGoogle Scholar
- Kawashima S, Shibata Y: Global existence and exponential stability of small solutions to nonlinear viscoelasticity. Communications in Mathematical Physics 1992,148(1):189-208. 10.1007/BF02102372MathSciNetView ArticleMATHGoogle Scholar
- Haraux A, Zuazua E: Decay estimates for some semilinear damped hyperbolic problems. Archive for Rational Mechanics and Analysis 1988,100(2):191-206. 10.1007/BF00282203MathSciNetView ArticleMATHGoogle Scholar
- Kopáčková M: Remarks on bounded solutions of a semilinear dissipative hyperbolic equation. Commentationes Mathematicae Universitatis Carolinae 1989,30(4):713-719.MathSciNetMATHGoogle Scholar
- Ball JM: Remarks on blow-up and nonexistence theorems for nonlinear evolution equations. The Quarterly Journal of Mathematics. Oxford 1977,28(112):473-486. 10.1093/qmath/28.4.473MathSciNetView ArticleMATHGoogle Scholar
- Yang Z: Existence and asymptotic behaviour of solutions for a class of quasi-linear evolution equations with non-linear damping and source terms. Mathematical Methods in the Applied Sciences 2002,25(10):795-814. 10.1002/mma.306MathSciNetView ArticleMATHGoogle Scholar
- Yang Z, Chen G: Global existence of solutions for quasi-linear wave equations with viscous damping. Journal of Mathematical Analysis and Applications 2003,285(2):604-618. 10.1016/S0022-247X(03)00448-7MathSciNetView ArticleMATHGoogle Scholar
- Zhijian Y: Initial boundary value problem for a class of non-linear strongly damped wave equations. Mathematical Methods in the Applied Sciences 2003,26(12):1047-1066. 10.1002/mma.412MathSciNetView ArticleMATHGoogle Scholar
- Yacheng L, Junsheng Z: Multidimensional viscoelasticity equations with nonlinear damping and source terms. Nonlinear Analysis: Theory, Methods & Applications 2004,56(6):851-865. 10.1016/j.na.2003.07.021MathSciNetView ArticleMATHGoogle Scholar
- Yang Z: Blowup of solutions for a class of non-linear evolution equations with non-linear damping and source terms. Mathematical Methods in the Applied Sciences 2002,25(10):825-833. 10.1002/mma.312MathSciNetView ArticleMATHGoogle Scholar
- Nakao M: A difference inequality and its application to nonlinear evolution equations. Journal of the Mathematical Society of Japan 1978,30(4):747-762. 10.2969/jmsj/03040747MathSciNetView ArticleMATHGoogle Scholar
- Ye Y: Existence of global solutions for some nonlinear hyperbolic equation with a nonlinear dissipative term. Journal of Zhengzhou University 1997,29(3):18-23.MathSciNetMATHGoogle Scholar
- Ye Y: On the decay of solutions for some nonlinear dissipative hyperbolic equations. Acta Mathematicae Applicatae Sinica. English Series 2004,20(1):93-100. 10.1007/s10255-004-0152-4MathSciNetView ArticleMATHGoogle Scholar
- Sattinger DH: On global solution of nonlinear hyperbolic equations. Archive for Rational Mechanics and Analysis 1968, 30: 148-172. 10.1007/BF00250942MathSciNetView ArticleMATHGoogle Scholar
- Komornik V: Exact Controllability and Stabilization, Research in Applied Mathematics. Masson, Paris, France; 1994:viii+156.Google Scholar
- Ye Y: Existence and nonexistence of global solutions of the initial-boundary value problem for some degenerate hyperbolic equation. Acta Mathematica Scientia 2005,25(4):703-709.MathSciNetMATHGoogle Scholar
- Gao H, Ma TF:Global solutions for a nonlinear wave equation with the -Laplacian operator. Electronic Journal of Qualitative Theory of Differential Equations 1999, (11):1-13.Google Scholar
- Adams RA: Sobolev Spaces, Pure and Applied Mathematics. Volume 6. Academic Press, New York, NY, USA; 1975:xviii+268.Google Scholar
- Ladyzhenskaya OA: The Boundary Value Problems of Mathematical Physics, Applied Mathematical Sciences. Volume 49. Springer, New York, NY, USA; 1985:xxx+322.View ArticleGoogle Scholar
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