Abstract
This chapter discusses the existence, linear and multilinear cases, approximation, and stability of abstract nonlinear wave equations. It is extremely useful to have an explicit expression for the functions. It is found that the nonlinearity is reproducing relative to the sequence. The idea is that much nonlinearity has computable expansion coefficients, when applied to a suitable complete orthonormal system. An explicit example is also given. The second remark concerns the case when the nonlinearity M(u) is the gradient of a functional where R denotes terms of higher order in h. This holds when M(u) is a cyclically monotone operator. The approximations have the form of a conservative Hamiltonian system of classical mechanics. The stability of the zero solution is investigated. The analysis makes essential use of the results of Rutkowski, who use the reproducing property of the nonlinearity to obtain explicit expansion coefficients.
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