Regular generalized solutions to semilinear wave equations

Hideo Deguchi; Michael Oberguggenberger

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arXiv · 1909.05705

Regular generalized solutions to semilinear wave equations

Hideo Deguchi·Michael Oberguggenberger

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Abstract

The paper is devoted to proving an existence and uniqueness result for generalized solutions to semilinear wave equations with a small nonlinearity in space dimensions 1, 2, 3. The setting is the one of Colombeau algebras of generalized functions. It is shown that for a nonlinearity of arbitrary growth and sign, but multiplied with a small parameter, the initial value problem for the semilinear wave equation has a unique solution in the Colombeau algebra of generalized functions of bounded type. The proof relies on a fixed point theorem in the ultra-metric topology on the algebras involved. In classical terms, the result says that the semilinear wave equations under consideration have global classical solutions up to a rapidly vanishing error.

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Hideo Deguchi, Michael Oberguggenberger. 2019-09-11. Regular generalized solutions to semilinear wave equations. https://arxiv.org/abs/1909.05705

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