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Generalized solutions to a characteristic Cauchy problem

Allaud E, Devoue V

Journal of Applied Analysis

2013

Vol. 19, nr 1

1--29

We give a meaning to the nonlinear characteristic Cauchy problem for the wave equation in base form by replacing it by a family of non-characteristic ones. This leads to a well-formulated problem in an appropriate algebra of generalized functions. We prove existence of a solution and we precise how it depends on the choice made. We also check that in the classical case (non-characteristic) our new solution coincides with the classical one.

Generalized solutions to a non Lipschitz- Cauchy problem

Devoue V.

Journal of Applied Analysis

2009

Vol. 15, nr 1

1-32

In this article we investigate solutions to a semilinear partial differential equation with non Lipschitz nonlinearity by using recent theories of generalized functions. To give a meaning to a non Lipschitz characteristic Cauchy problem with irregular data, we replace it by a three parameter family of problems. The first parameter turns the problem into a family of Lipschitz problems, the second one converts the given problem to a non-characteristic one, whereas the third one regularizes the data. Finally, the family of problems is solved in an appropriate three parametric (C,Ε,P) algebra.