Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-17T20:09:29.075Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak asymptotic decay via a “relaxed invariance principle” for a wave equation with nonlinear, non-monotone damping

Published online by Cambridge University Press:  14 November 2011

M. Slemrod
M. Slemrod
Affiliation:
Center for Mathematical Sciences, University of Wisconsin, Madison, WI 53705, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

This paper considers the problem of asymptotic decay as t → ∞ of solutions of the wave equation utt − Δu = −a(x)β(ut, ∇u), (t,x) ∊ ℝ+ × Ω (a bounded, open, connected set in ℝN, N≧ 1, with smooth boundary), u =0 on ℝ+ × ∂Ω. The nonlinear function β is not assumed to be globally Lipschitz continuous, β(0, y2, …, yN+1) = 0, y1 β(y1…,yN+1) ≧ 0 for all y ∊ ℝN+1; β is not assumed to be monotone in y1. Under additional restrictions on the kernel of β, conditions are given which imply that [u, ut,] converges to [0,0] weakly in H = H10(Ω) × L2(Ω) as t → ∞. The work generalises earlier results of Dafermos and Haraux where strong decay in H as t → ∞ was obtained in the case β(y1 …, yN+1) = q(y1), q monotone on ℝ.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 113 , Issue 1-2 , 1989 , pp. 87 - 97
Copyright
Copyright © Royal Society of Edinburgh 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Artstein, Z.. The limiting equations of nonautonomous ordinary differential equations. J. Differential Equations 25 (1977), 184–201.CrossRefGoogle Scholar
2Ball, J. M.. Strongly continuous semigroups, weak solutions, and the variation of constants formula. Proc. Am. Math. Soc. 63 (1977), 370–373.Google Scholar
3Ball, J. M.. On the asymptotic behavior of generalized processes, with applications to nonlinear evolution equations. J. Differential Equations 27 (1978), 224–265.CrossRefGoogle Scholar
4Ball, J. M. and Slemrod, M.. Feedback stabilization of distributed semilinear control systems. Appl. Math. Optim. 5 (1979), 169–179.CrossRefGoogle Scholar
5Ball, J. M. and Slemrod, M.. Nonharmonic Fourier series and the stabilization of distributed semilinear control systems. Comm. Pure Appl. Math. 32 (1979), 555–587.CrossRefGoogle Scholar
6Brezis, H.. Asymptotic behavior of some evolutionaty systems. In Nonlinear Evolution Equations, ed. Crandall, M. G., pp. 141–154 (New York; Academic Press, 1978).Google Scholar
7Chafee, N. and Infante, E. F.. A bifurcation problem for a nonlinear partial differential equation of parabolic type. Applicable Anal. 4 (1974), 17–37.CrossRefGoogle Scholar
8Dafermos, C. M.. Asymptotic behavior of solutions of evolution equations. In Nonlinear Evolution Equations, ed. Crandall, M. G., pp. 103–123 (New York: Academic Press, 1978).Google Scholar
9Dafermos, C. M. and Slemrod, M.. Asymptotic behavior of nonlinear contraction semigroups. J. Functional Anal. 13 (1973), 97–106.CrossRefGoogle Scholar
10DiPerna, R. J.. Measure-valued solutions to conservation laws. Arch. Rational Mech. Anal. 88 (1985), 223–270.CrossRefGoogle Scholar
11Hale, J. K.. Dynamical systems and stability. J. Math. Anal. Appl. 26 (1969), 39–59.CrossRefGoogle Scholar
12Hale, J. K.. Asymptotic Behavior of Dissipative Systems (Providence, RI: American Mathematical Society, 1988).Google Scholar
13Hale, J. K. and Infante, E. F.. Extended dynamical systems and stability theory. Proc. Nat. Acad. Sci. U.S.A. 58 (1967), 405–409.CrossRefGoogle ScholarPubMed
14Haraux, A.. Stabilization of trajectories for some weakly damped hyperbolic equations. J. Differential Equations 59 (1985), 145–154.CrossRefGoogle Scholar
15Haraux, A.. Nonlinear Evolution Equations: Global Behavior of Solution, Lecture Notes in Math. 841 (Berlin: Springer, 1981).CrossRefGoogle Scholar
16Henry, D.. Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840 (Berlin: Springer, 1981).CrossRefGoogle Scholar
17LaSalle, J. P.. The extent of asymptotic stability. Proc. Nat. Acad. Sci. U.S.A. 46 (1960), 363–365.CrossRefGoogle ScholarPubMed
18LaSalle, J. P. (with appendix by Z. Artstein). The Stability of Dynamical Systems, SIAM Regional Conf. Ser. in Appl. Math. 25 (Philadelphia: SIAM Publications, 1976).CrossRefGoogle Scholar
19Pavel, N. H.. Nonlinear Evolution Operators and Semigroups, Lecture Notes in Math. 1260 (Berlin: Springer, 1987).CrossRefGoogle Scholar
20Pazy, A.. A class of semilinear equations of evolution. Israel J. Math. 20 (1975), 23–36.CrossRefGoogle Scholar
21Pazy, A.. Semigroups of Linear Operators and Applications to Partial Differential Equations (Berlin: Springer, 1983).CrossRefGoogle Scholar
22Saperstone, S. H.. Semidynamical Systems in Infinite Dimensional Spaces, Appl. Math. Sci. 37 (Berlin: Springer, 1981).CrossRefGoogle Scholar
23Schonbek, M. E.. Convergence of solutions to nonlinear dispersive equations. Comm. Partial Differential Equations 7 (1982), 959–1000.Google Scholar
24Slemrod, M.. Asymptotic behavior of a class of abstract dynamical systems. J. Differential Equations 7 (1970), 584–600.CrossRefGoogle Scholar
25Tartar, L.. Compensated compactness and applications to partial differential equations. In Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, vol. 4, ed Knops, R. J., Res. Notes in Math. 39, pp. 136–211 (London: Pitman, 1975).Google Scholar
26Walker, J. A.. Dynamical Systems and Evolution Equations (New York: Plenum, 1980).CrossRefGoogle Scholar
27Webb, G. F.. Compactness of bounded trajectories of dynamical systems in infinite dimensional spaces. Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), 19–33.CrossRefGoogle Scholar
28Young, L. C.. Lectures on the Calculus of Variations and Optimal Control Theory (Philadelphia: W. B. Saunders, 1969).Google Scholar
29Zubov, V. I.. Methods of A. M. Liapunov and their Applications (translation) (Noordhoff: P. Groningen, 1964).Google Scholar