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

  • Jürgen Weyer (a1)

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.

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1.Bazley, N. W. and Küpper, T.. Branches of solutions in nonlinear eigenvalue problems. Applications of nonlinear analysis in physical sciences (Boston: Pitman, 1981).
2Bazley, N. W. and Küpper, T.. Bifurcation diagrams and periodic solutions to nonlinear equations. Z. Angew. Math. Phys. 32 (1981), 470473.
3Bazley, N. W. and Weinacht, R. J.. A class of operator equations in nonlinear mechanics. Applicable Anal, to appear.
4Browder, F. E.. On non-linear wave equations. Math. Z. 80 (1962), 249264.
5Brüll, L.. Zur Existenztheorie von nichtlinearen Schrödinger- und Wellengleichungen ohne Lipschitz-Voraussetzungen. (Cologne University, doctoral thesis, 1982).
6Dickey, R. W.. Free vibrations and dynamic buckling of the extensible beam. J. Math. Anal. Appl. 29 (1970), 443454.
7Heinz, E. and Wahl, W.v.. Browder, Zu einem Satz von F. E.über nichtlineare Wellengleichungen. Math. Z. 141 (1975), 3345.
8Kato, T.. Perturbation theory for linear operators, (Berlin: Springer, 1980).
9Lions, J. L.. On some questions in boundary value problems in mathematical physics (Rio de Janeiro: Lecture notes UFRJ, 1977).
10Medeiros, L. A.. On a new class of nonlinear wave equations. J. Math. Anal. Appl. 69 (1979), 252262.
11Nayfeh, A. H. and Mook, D. T.. Nonlinear oscillations (New York: Wiley-Interscience, 1979).
12Vainberg, M. M.. Variational methods for the study of nonlinear operators (San Francisco: Holden-Day, 1964).
13Weyer, J.. Über maximale Monotonie von Operatoren des Typs L*ΦºL Manuscripta Math. 28 (1979), 305316.
14Weyer, J.. Nonlinear elliptic equations and operators of the type L*ΦºL. Nonlinear Anal. (TMA) 6 (1982), 615623.
15Weyer, J.. Maximal monotonicity of operators with sufficiently large domain and application to the Hartree problem. Manuscripta Math. 38 (1982), 163174.

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