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Stability theory for quasi-linear wave equations with linear damping

Published online by Cambridge University Press:  14 November 2011

R. W. Dickey
R. W. Dickey
Affiliation:
University of Wisconsin, Madison, U.S.A.
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Synopsis

Large time behaviour of solutions to a damped quasi-linear wave equation are studied. Conditions are obtained which guarantee the global existence of a classical solution. The asymptotic behaviour of this solution is studied in the case of a unique equilibrium solution and in the case of multiple equilibria. The results are applied to various special examples.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 88 , Issue 1-2 , 1981 , pp. 25 - 41
Copyright
Copyright © Royal Society of Edinburgh 1981

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References

1Courant, R. and Hilbert, D.. Methods of Mathematical Physics, Vol. I (New York: Interscience, 1953).Google Scholar
2Chafee, N. and Infante, E. F.. A bifurcation problem for a nonlinear partial differential equation of parabolic type. Applicable Anal. 4 (1974), 17–37.Google Scholar
3Webb, G. F.. A bifurcation problem for a nonlinear hyperbolic partial differential equation. SIAM J. Math. Anal. 10 (1979), 922–932.Google Scholar
4Dickey, R. W., Asymptotic behavior of solutions of hyperbolic, semi-linear partial differential equations. J. Math. Anal. Appl. 55 (1976), 380–393.Google Scholar
5Zabusky, N. J.. Exact solution for the vibrations of a nonlinear continuous model string. J. Mathematical Phys. 3 (1962), 1028–1039.Google Scholar
6Lax, P. D.. Development of singularities of solutions of nonlinear hyperbolic partial differential equations. J. Mathematical Phys. 5 (1964), 611–613.Google Scholar
7Courant, R. and Hilbert, D.. Methods of Mathematical Physics, Vol. 2 (New York: Interscience, 1962).Google Scholar
8Coddington, E. A. and Levinson, N.. Theory of Ordinary Differential Equations (New York: McGraw-Hill, 1955).Google Scholar
9Ince, E. L.. Ordinary Differential liquations (New York: Dover, 1956).Google Scholar
10Dickey, R. W.. A quasi-linear evolution equation and the method of Galerkin. Proc. Amer. Math. Soc. 37 (1973), 149–156.Google Scholar