Propagation of singularities for generalized solutions to nonlinear wave equations

Abstract

The paper is devoted to regularity theory of generalized solutions to semilinear wave equations with a small nonlinearity. The setting is the one of Colombeau algebras of generalized functions. It is shown that in one space dimension, an initial singularity at the origin propagates along the characteristic lines emanating from the origin, as in the linear case. The proof is based on a fixed point theorem in a suitable ultrametric topology on the subset of Colombeau solutions possessing the required regularity. The paper takes up the initiating research of the 1970s on anomalous singularities in classical solutions to semilinear hyperbolic equations, and shows that the same behavior is attained in the case of non-classical, generalized solutions.

New fixed point theorems andcharacterization of the continuity inColombeau algebra

“…This will allow us to use the powerful tools from this theory to combine generalized function data with nonlinearities and measure regularity. We gives some properties in the theory of tpoplogical C-modules and locally convex topological C-modules, and illustrated this with an application the evolution propblem.In [5] the authors, shows that in one space dimension, an initial singularity at the origin propagates along the characteristic lines emanating from the origin, as in the linear case. The proof is based on a fixed point theorem in a suitable ultrametric topology on the subset of Colombeau solutions possessing the required regularity.…”

Regular generalized solutions to semilinear wave equations

“…We present a new existence result of a global generalized solution without growth or sign restrictions on the nonlinearity f , for initial data possessing so-called G 0 -regularity. It is motivated by a result on propagation of singularities in the onedimensional case of the authors [4]. Our main tool will be the Banach fixed point theorem in the so-called sharp topology, a complete ultra-metric topology on the Colombeau algebras.…”