Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-24T22:37:36.626Z Has data issue: false hasContentIssue false
  • English
  • Français

Regular solutions of wave equations containing nonlinearities of the type L Φ L

Published online by Cambridge University Press:  14 November 2011

Jürgen Weyer
Jürgen Weyer
Affiliation:
Departamento Matemática, Universidad de Santiago de Chile, Santiago, Chile
Get access Rights & Permissions [Opens in a new window]

Synopsis

Regular solutions of the forced nonlinear wave equation uu + L4u + LΦLu = r are studied in Hilbert spaces. L is a linear, positive, selfadjoint operator and the nonlinear nucleus Φ(u) = f(|u|2)u is generated by a C1-function f, such that LΦ(Lu) = f(|Lu|2)L2u. If the initial value data u(0) = ϕ and u1(0) = ψ belong to the domain D(Lk+4) and D(Lk+2), respectively, and if rεD(Lk), then there is a (global) solution u(t) such that u ε D(Lk+4), ut ε D(Lk+2) and uuε D(Lk) for all times t. The abstract result is applied to examples in nonlinear elasticity theory.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 95 , Issue 1-2 , 1983 , pp. 147 - 152
Copyright
Copyright © Royal Society of Edinburgh 1983

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

1.Bazley, N. W. and Küpper, T.. Branches of solutions in nonlinear eigenvalue problems. Applications of nonlinear analysis in physical sciences (Boston: Pitman, 1981).Google Scholar
2Bazley, N. W. and Küpper, T.. Bifurcation diagrams and periodic solutions to nonlinear equations. Z. Angew. Math. Phys. 32 (1981), 470–473.CrossRefGoogle Scholar
3Bazley, N. W. and Weinacht, R. J.. A class of operator equations in nonlinear mechanics. Applicable Anal, to appear.Google Scholar
4Browder, F. E.. On non-linear wave equations. Math. Z. 80 (1962), 249–264.CrossRefGoogle Scholar
5Brüll, L.. Zur Existenztheorie von nichtlinearen Schrödinger- und Wellengleichungen ohne Lipschitz-Voraussetzungen. (Cologne University, doctoral thesis, 1982).Google Scholar
6Dickey, R. W.. Free vibrations and dynamic buckling of the extensible beam. J. Math. Anal. Appl. 29 (1970), 443–454.CrossRefGoogle Scholar
7Heinz, E. and Wahl, W.v.. Browder, Zu einem Satz von F. E.über nichtlineare Wellengleichungen. Math. Z. 141 (1975), 33–45.CrossRefGoogle Scholar
8Kato, T.. Perturbation theory for linear operators, (Berlin: Springer, 1980).Google Scholar
9Lions, J. L.. On some questions in boundary value problems in mathematical physics (Rio de Janeiro: Lecture notes UFRJ, 1977).Google Scholar
10Medeiros, L. A.. On a new class of nonlinear wave equations. J. Math. Anal. Appl. 69 (1979), 252–262.CrossRefGoogle Scholar
11Nayfeh, A. H. and Mook, D. T.. Nonlinear oscillations (New York: Wiley-Interscience, 1979).Google Scholar
12Vainberg, M. M.. Variational methods for the study of nonlinear operators (San Francisco: Holden-Day, 1964).Google Scholar
13Weyer, J.. Über maximale Monotonie von Operatoren des Typs L*ΦºL Manuscripta Math. 28 (1979), 305–316.CrossRefGoogle Scholar
14Weyer, J.. Nonlinear elliptic equations and operators of the type L*ΦºL. Nonlinear Anal. (TMA) 6 (1982), 615–623.CrossRefGoogle Scholar
15Weyer, J.. Maximal monotonicity of operators with sufficiently large domain and application to the Hartree problem. Manuscripta Math. 38 (1982), 163–174.CrossRefGoogle Scholar