- Research Article
- Open Access
Exponential Decay of Energy for Some Nonlinear Hyperbolic Equations with Strong Dissipation
Advances in Difference Equations volume 2010, Article number: 357404 (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.
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 [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
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.
The solution is called the weak solution of the problem (1.1)–(1.3) on , if satisfy
and in , in .
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.
Let , then , and the inequality holds with a constant depending on , and , provided that, if and .
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 .
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
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.
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 .
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.
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
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 ).
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).
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
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.
Andrews G:On the existence of solutions to the equation . Journal of Differential Equations 1980,35(2):200-231. 10.1016/0022-0396(80)90040-6
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)90019-5
Ang DD, Dinh PN: Strong solutions of quasilinear wave equation with non-linear damping. SIAM Journal on Mathematical Analysis 1985, 19: 337-347. 10.1137/0519024
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/BF02102372
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/BF00282203
Kopáčková M: Remarks on bounded solutions of a semilinear dissipative hyperbolic equation. Commentationes Mathematicae Universitatis Carolinae 1989,30(4):713-719.
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/28.4.473
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.306
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)00448-7
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.412
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.2003.07.021
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.312
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/03040747
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.
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/s10255-004-0152-4
Sattinger DH: On global solution of nonlinear hyperbolic equations. Archive for Rational Mechanics and Analysis 1968, 30: 148-172. 10.1007/BF00250942
Komornik V: Exact Controllability and Stabilization, Research in Applied Mathematics. Masson, Paris, France; 1994:viii+156.
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.
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.
Adams RA: Sobolev Spaces, Pure and Applied Mathematics. Volume 6. Academic Press, New York, NY, USA; 1975:xviii+268.
Ladyzhenskaya OA: The Boundary Value Problems of Mathematical Physics, Applied Mathematical Sciences. Volume 49. Springer, New York, NY, USA; 1985:xxx+322.
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).