Research Article | Open | Published:
Global Existence and Asymptotic Behavior of Solutions for Some Nonlinear Hyperbolic Equation
Journal of Inequalities and Applicationsvolume 2010, Article number: 895121 (2010)
The initial boundary value problem for a class of hyperbolic equation with nonlinear 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 show the asymptotic behavior of the global solutions through the use of an important lemma of Komornik.
We are concerned with the global solvability and 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 .
The interaction between the damping and the source terms was first considered by Levine [8, 9] in the case . He showed that solutions with negative initial energy blow up in finite time. Georgiev and Todorova  extended Levine's result to the nonlinear damping case . In their work, the authors considered (1.1)–(1.3) with and introduced a method different from the one known as the concavity method. They determined suitable relations between and , for which there is global existence or alternatively finite time blow-up. Precisely, they showed that solutions with negative energy continue to exist globally in time if and blow up in finite time if and the initial energy is sufficiently negative. Vitillaro  extended these results to situations where the damping is nonlinear and the solution has positive initial energy. For the Cauchy problem of (1.1), Todorova  has also established similar results.
Zhijian in [13–15] 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 is greatly different from the Laplace operator . By mean of the Galerkin method and compactness criteria and a difference inequality introduced by Nakao , the author [19, 20] 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 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 let denote the closure in of . For simplicity of notation, hereafter we denote by the Lebesgue space norm, and 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. Main Results
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.
For latter applications, we list up some lemmas.
Let , then and the inequality holds with a constant depending on , and , provided that (i) if ; (ii) , .
Lemma 2.2 (see ).
Let be a nonincreasing function and assume that there are two constants and such that
then , for all , if , and , for all , if , where and are positive constants independent of .
Let be a solutions to problem (1.1)–(1.3). Then is a nonincreasing function for and
By multiplying (1.1) by and integrating over , we get
Therefore, is a nonincreasing function on .
Suppose that , and , . If , , then there exists such that the problem (1.1)–(1.3) has a unique local solution in the class
Assume that the hypotheses in Theorem 2.4 hold, then
By the definition of and , we have the following identity:
Since , so we have . Therefore, we obtain from (2.9) that
Suppose that and . If and such that
then , for each .
Since , so . Then there exists such that for all . Thus, we get from (2.3) and (2.8) that
and it follows from Lemma 2.3 that
Next, we easily arrive at from Lemma 2.1, (2.11), and (2.13) that
which implies that , for all . By noting that
we repeat the steps (2.12)–(2.14) to extend to . By continuing the procedure, the assertion of Lemma 2.6 is proved.
Assume that , and , . is a local solution of problem (1.1)–(1.3) on . If and satisfy (2.11), 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 2.7, we get from Lemma 2.6 that on . So the formula (2.8) in Lemma 2.5 holds on . Therefore, we have from (2.8) and Lemma 2.3 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).
The following theorem shows the asymptotic behavior of global solutions of problem (1.1)–(1.3).
If the hypotheses in Theorem 2.7 are valid, and , and , , then the global solutions of problem (1.1)–(1.3) have the following asymptotic behavior:
Multiplying by on both sides of (1.1) and integrating over , we obtain that
so, substituting the formula (2.21) into the right-hand side of (2.20), we get that
We obtain from (2.14) and (2.12) that
It follows from (2.22), (2.23), and (2.24) that
We have from Hölder inequality, Lemma 2.1, and (2.17) that
and similarly, we have
Substituting the estimates (2.26) and (2.27) into (2.25), we conclude that
It follows from that .
We get from Young inequality and Lemma 2.3 that
From Young inequality, Lemmas 2.1 and 2.3, and (2.17), We receive that
Choosing small enough and , such that
then, substituting (2.29) and (2.30) into (2.28), we get
Therefore, we have from Lemma 2.2 that
where is a positive constant depending on .
We conclude from (2.17) and (2.33) that and
The proof of Theorem 2.8 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.
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
Kopackova 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 1977, 28(112):473–486.
Levine HA: Instability and nonexistence of global solutions to nonlinear wave equations of the form . Transactions of the American Mathematical Society 1974, 192: 1–21.
Levine HA: Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM Journal on Mathematical Analysis 1974, 5: 138–146. 10.1137/0505015
Georgiev V, Todorova G: Existence of a solution of the wave equation with nonlinear damping and source terms. Journal of Differential Equations 1994, 109(2):295–308. 10.1006/jdeq.1994.1051
Vitillaro E: Global nonexistence theorems for a class of evolution equations with dissipation. Archive for Rational Mechanics and Analysis 1999, 149(2):155–182. 10.1007/s002050050171
Todorova G: Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms. Journal of Mathematical Analysis and Applications 1999, 239(2):213–226. 10.1006/jmaa.1999.6528
Zhijian Y: 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
Zhijian Y, 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
Liu YC, Zhao JS: 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. Natural Science Edition 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.
Sattinger DH: On global solution of nonlinear hyperbolic equations. Archive for Rational Mechanics and Analysis 1968, 30: 148–172.
Komornik V: Exact Controllability and Stabilization, The Multiplier Method, Research in Applied Mathematics. Masson, Paris, France; 1994:viii+156.
This Research was supported by the Natural Science Foundation of Henan Province (no. 200711013), The Science and Research Project of Zhejiang Province Education Commission (no. Y200803804), 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 (2008–2012).