Approximation of wave equations with reproducing nonlinearities

Abstract

Here A is a positive definite, linear, self-adjoint operator with domain D, in a separable Hilbert space X. The nonlinear operator M(u) has a linear domain D, satisfying 2 = D, n D, The question of existence and uniqueness for solutions of (1) goes back to Jijrgens [l], who considered a particular equation of importance in quantum field theory. Browder [2] obtained a completely operator theoretical abstract existence theorem, and his results have recently been extended by Heinz and v. Wahl [3], whose work plays an essential role in our presentation. See also the book of Reed [4]. Here we consider the approximation of solutions of (1) by the method of Faedo-Galerkin, which we rediscovered in our analysis. A detailed treatment of this method is given in the book of Lions [5], where it is used to prove the existence of weak solutions. Here, however, we consider the method from the viewpoint of the numerical and analytical approximation of (1). Our essential idea is to introduce nonlinearities M which are “reproducing” relative to a complete orthonormal sequence (C.O.S.) {ui}y, and thus obtain a finite system of explicitly known FaedoGalerkin approximating ordinary differential equations. This system can be handled by known methods of numerical integration, Lyapunov theory, etc. We show that if {t+jy are eigenfunctions of A, then the solution of the Faedo-Galerkin approximations converges to that of (1) under the assumptions of [3]. Our procedure can bc considered as a generalized separation of variables for a class of nonlinear wave equations. It extends to inhomogeneous equations U” + Au + M(u) = f; as well as parabolic equations u’ + Au + M(u) = jI The concept of a reproducing nonlinearity was first considered in [6], in the approximation of Ljusternik-Schnirelmann critical values, and was also applied to nonlinear wave equations in [7]. Solutions of the Faedo-Galerkin type were also used by Dickey [8] in the analysis of the extensible beam.