Regularization for heat kernel in nonlinear parabolic equations

Abstract

We prove existence-uniqueness theorems for some kinds of non-linear parabolic equations (cf. [2, 3, 15]) with singular initial data and non-Lipschitz's nonlinearities in a framework of Colombeau's algebras using different kinds of regularization for singularities appearing in the equations. We establish the convergence of a family of regularized solutions to the classical solutions (if they exist), when nonlinear term g(u) is of Lipschitz's class and ε → 0. Moreover, we find solutions not available in classical approach.