On the Existence and Uniqueness of Mild and Strong Solutions of a Generalized Nonlinear Heat Equation

Abstract

In this paper, we study well-posedness of the Cauchy problem for a nonlinear generalized heat equation with initial data in Besov and Triebel{Lizorkin spaces. We prove existence and uniqueness of mild and strong solutions which are local in time. The crucial point is the use of estimates and mapping properties of the generalized Gauss{Weierstrass semigroup in function spaces under consideration. Moreover, we study regularity properties of solutions with respect to space and time.