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On a quasilinear wave equation with nonlinear damping

Published online by Cambridge University Press:  14 November 2011

Dang Dinh Hai
Dang Dinh Hai
Affiliation:
Department of Mathematics, Dai Hoc Tong Hop, Ho Chi Minh City University, Ho Chi Minh, Vietnam
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Synopsis

We prove the global existence and uniqueness of the solution of the initial and boundary value problem for the equation

by using the classical Galerkin method when the forcing term and the initial data are in some sense small. The asymptotic behaviour of the solution as t → ∞ is also considered.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 110 , Issue 3-4 , 1988 , pp. 227 - 239
Copyright
Copyright © Royal Society of Edinburgh 1988

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References

1Andrews, G.. On the existence of solutions to the equation u tt = u xxt + σ(u x)x. J. Differential Equations 35 (1980), 200–231.CrossRefGoogle Scholar
2Andrews, G. and Ball, J.. Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity. J. Differential Equations 44 (1982), 306–341.CrossRefGoogle Scholar
3Clements, J. C.. On the existence and uniqueness of solutions of the equation ΔNut = f. Canad. Math. Bull. 18 (1975), 181–187.CrossRefGoogle Scholar
4Dafermos, C. M.. The mixed initial boundary value problem for the equation of nonlinear one-dimensional viscoelasticity. J. Differential Equations 6 (1969), 71–86.CrossRefGoogle Scholar
5Hai, Dang Dinh. On a strongly damped quasilinear wave equation. Demonstratio Math. 19 (1986), 327–340.Google Scholar
6Greenberg, J. M.. On the existence, uniqueness and stability of solutions of the equation σ0u tt, = E(X X)X XX + λX xxt. J. Math. Anal. Appl. 25 (1969), 575–591.CrossRefGoogle Scholar
7Greenberg, J. M., MacCamy, R. C. and Mizel, V. J.. On the existence, uniqueness and stability of solutions of the equation σ(u x)u xx + λX xtx = ρ0u tt. J. Math. Mech. 17 (1968), 707–728.Google Scholar
8Prestel, M.. Forced oscillations for the solutions of nonlinear hyperbolic equation. Nonlinear Anal. 6 (1982), 209–216.CrossRefGoogle Scholar
9Yamada, Y.. Quasilinear wave equations and related nonlinear evolution equation. Nagoya Math. J. 84 (1981), 31–83.CrossRefGoogle Scholar