Generalized solutions in the Egorov formulation of nonlinear diffusion equations and linear hyperbolic equations in the Colombeau algebra

Abstract

We investigate generalized solutions of nonlinear diffusion equations and linear hyperbolic equations with discontinuous coefficients in the framework of Colombeau’s algebra of generalized functions. Under Egorov’s formulation, we obtain results on existence and uniqueness of generalized solutions, which are shown to be consistent with classical solutions. The example of a linear hyperbolic equation given by Hurd and Sattinger [8] has no distributional solutions in Schwartz’s sense, but has the unique generalized solution. We study what distribution is associated with it, namely, how it behaves on the level of information of distribution theory.