Generalized solutions to a non Lipschitz Goursat problem

Victor Devoue (1)

1 AOC - Analyse Optimisation Controle

In this paper we investigate solutions to the semi-linear wave equation in canonical form with non Lipschitz non-linearity and distributions or other generalized functions as data. To give a meaning to the Goursat problem with irregular data, we replace it by a biparametric family of problems. The first parameter turns the problem into a family of Lipschitz problems, the second one regularizes the data. Finally, the problem is solved in an appropriate algebra. We show that the solution is equal to the non-regularized one. In the examples, we take advantage of our results to give a new approach of the blow-up problem.

Mots clés: non-linear partial differential equation; wave equation; Goursat problem; regularization of problems; algebras of generalized functions.; algebras of generalized functions

Domaines: Equations aux dérivées partielles [math.AP]

Victor Devoue. Generalized solutions to a non Lipschitz Goursat problem. Differential Equations and Applications, 2009, 1 (2), pp.153-178. ⟨hal-00345090v2⟩