Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-12T19:46:44.375Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized quasilinear hyperbolic equations and Yosida approximations

Part of: Hyperbolic equations and systems

Published online by Cambridge University Press:  09 April 2009

Jong Yeoul Park ,
Il Hyo Jung  and
Yong Han Kang
Yong Han Kang
Affiliation:
Department of Mathematics Pusan National UniversityPusan 609–735Korea e-mail: [email protected], [email protected]
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.

We will show the existence, uniqueness and regularity of global solutions for the Cauchy problem for nonlinear evolution equations with the damping term .

As an application of our results, we give the global solvability and regularity of the mixed problem with Dirichiet boundary conditions:

Keywords

Yosida approximationsquasilinear hyperbolic equationglobal solutionsregularity

MSC classification

Secondary: 35L70: Nonlinear second-order hyperbolic equations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 74 , Issue 1 , February 2003 , pp. 69 - 86
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Brito, E. H., ‘The damped elastic stretched string equation generalized: existence, uniqueness, regularity and stability’, Applicable Anal. 13 (1982), 219–233.CrossRefGoogle Scholar
[2]Dickey, R. W., ‘Infinite systems of nonlinear oscillation equations related to the string’, Proc. Amer Math. Soc. 23 (1969), 459–648.CrossRefGoogle Scholar
[3]Dix, J. and Torrejón, R., ‘A quasilinear integrodifferential equation of hyperbolic type’, Differential Integral Equations 6 (1993), 431–447.CrossRefGoogle Scholar
[4]Ikehata, R. and Okazawa, N., ‘Yosida approximation and nonlinear hyperbolic equation’, Nonlinear Anal. 15 (1990), 479–495.CrossRefGoogle Scholar
[5]Lakshmikantham, V., Leela, S. and Martynyuk, A. A., Stability analysis of nonlinear systems (Marcel Dekker, New York, 1989).Google Scholar
[6]Menzala, G. P., On classical solutions of a quasilinear hyperbolic equation, Memórias de Matemática da Universidade (Federal do Rio de Janeiro, 1978).Google Scholar
[7]Narasimha, R., ‘Nonlinear vibration of an elastic string’, J. Sound Vibration 8 (1968), 134–146.CrossRefGoogle Scholar
[8]Park, J. Y. and Jung, I. H., ‘On a class of quasilinear hyperbolic equations and Yosida approximations’, Indian J. Pure Appl. Math. 30 (1999), 1091–1106.Google Scholar
[9]Park, J. Y., Jung, I. H. and Kang, Y. H., ‘Some quasilinear hyperbolic equations and Yosida approximations’, Bull. Korean Math. Soc. 38 (2001), 505–516.Google Scholar
[10]Pohozaev, S. I., ‘On a class of quasilinear hyperbolic equations’, Mat. Sb. 25 (1975), 145–158.CrossRefGoogle Scholar
[11]Tanabe, H., Equations of evolution (Pitman, London, 1987).Google Scholar
[12]Yamada, Y., ‘On some quasilinear wave equations with dissipative terms’, Nagoya Math. J. 97 (1982), 17–39.CrossRefGoogle Scholar