Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-02T22:57:33.719Z Has data issue: false hasContentIssue false

On periodic solutions of a semilinear hyperbolic parabolic equation

Published online by Cambridge University Press:  17 April 2009

Mitsuhiro Nakao
Affiliation:
Department of Mathematics College of General Education, Kyushu University, Fukuoka 810, Japan
Hisako Kato
Affiliation:
Department of Mathematics College of General Education, Kyushu University, Fukuoka 810, Japan
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Uniqueness and regularity of periodic solutions to the semilinear dissipative wave equation with small parameter ∈ > 0,

are investigated when g(u) has a certain ‘critical’ nonlinearity.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Clements, J.C., ‘Existence theorem for some nonlinear equation of evolution’, Canad. J. Math. 22 (1970), 726745.CrossRefGoogle Scholar
[2]Kato, H. and Nakao, M., ‘Existence of strong and smooth periodic solutions of some nonlinear evolution equations’, Math. Rep. Kyushu Univ. 14 (1983), 5788.Google Scholar
[3]Milani, A.J., ‘Time periodic smooth solutions of hyperbolic quasi-linear equations with dissipation term and their approximation by parabolic equations’, Ann. Mat. Pura Appl. 140 (1985), 331344.CrossRefGoogle Scholar
[4]Nakao, M., ‘Bounded, periodic and almost periodic solutions of nonlinear hyperbolic partial differential equations’, J. Differential Equations 23 (1977), 368386.CrossRefGoogle Scholar
[5]Nakao, M., ‘Global existence of classical solutions to the initial-boundary value problem of the semilinear wave equations with a degenerate dissipative term’, Nonlinear Anal. 14 (1990), 115140.CrossRefGoogle Scholar
[6]Rabinowitz, P.H., ‘Periodic solutions of nonlinear hyperbolic partial differential equations’, Comm. Pure. Appl. Math. 20 (1967), 145205.CrossRefGoogle Scholar
[7]Sather, J., ‘The existence of a global classical solution of the initial-boundary value problem for □u + u 3 = f’, Arch. Rational Mech. Anal. 22 (1966), 292307.CrossRefGoogle Scholar
[8]Vejvoda, O., Partial differential equations: time-periodic solutions (Martinus Nijhoff Publishers, The Hague, Boston, London, 1982).CrossRefGoogle Scholar
[9]von Wahl, W., ‘Klassische Lösungen nichtlinearer Wellengleichungen im Grossen’, Math. Z. 112 (1969), 241279.CrossRefGoogle Scholar
[10]von Wahl, W., ‘Periodic solutions of nonlinear wave equations with a dissipative term’, Math. Ann. 190 (1971), 313322.CrossRefGoogle Scholar