2019/09/11 by Hideo Deguchi, Deguchi, Hideo, Michael Oberguggenberger +1
Mathematics · #35D05 #35D10 #46F30 (Primary) 35L71 (Secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.1909.05705
openalex publication_date 2019/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.