Author:
Deguchi Hideo,Oberguggenberger Michael
Abstract
AbstractThe 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.
Funder
University of Innsbruck and Medical University of Innsbruck
Publisher
Springer Science and Business Media LLC
Reference33 articles.
1. Biagioni, H.A.: A nonlinear theory of generalized functions. Lecture Notes in Mathematics, vol. 1421. Springer, Berlin (1990)
2. Colombeau, J.-F.: New generalized functions and multiplication of distributions. North-Holland Mathematics Studies, vol. 84. North-Holland, Amsterdam (1984)
3. Colombeau, J.-F.: Elementary introduction to new generalized functions. North-Holland Mathematics Studies, vol. 113. North-Holland, Amsterdam (1985)
4. Deguchi, H., Oberguggenberger, M.: Propagation of singularities for generalized solutions to nonlinear wave equations. J. Fixed Point Theory Appl. 22, 67 (2020)
5. Delcroix, A.: Regular rapidly decreasing nonlinear generalized functions. Application to microlocal regularity. J. Math. Anal. Appl. 327, 564–584 (2007)