Generalized Solutions of a Nonlinear Parabolic Equation with Generalized Functions as Initial Data

TL;DR

This paper extends the existence of solutions for nonlinear parabolic equations to initial data given by any Colombeau generalized function, broadening the class of initial conditions for which solutions are guaranteed.

Contribution

It generalizes previous results by allowing any Colombeau generalized function as initial data, using recent algebraic and topological advances in Colombeau theory.

Findings

Proves existence of solutions with generalized initial data

Extends previous results to broader class of initial functions

Utilizes recent developments in Colombeau algebra and topology

Abstract

In \cite{bf} Br\'ezis and Friedman prove that certain nonlinear parabolic equations, with the $\delta$-measure as initial data, have no solution. However in \cite{cl} Colombeau and Langlais prove that these equations have a unique solution even if the $\delta$-measure is substituted by any Colombeau generalized function of compact support. Here we generalize Colombeau and Langlais their result proving that we may take any generalized function as the initial data. Our approach relies on resent algebraic and topological developments of the theory of Colombeau generalized functions and results from \cite{A}.