Weak Solutions of Parabolic Equations in Whole Space

Abstract

This chapter explains how one can justify theoretically the formal manipulations of Chap. 1 We introduce precisely what is the concept of weak solutions and justify the use of integrations by parts. We show that the concept of solutions in the sense of distributions has good stability properties and that intuitive conservation laws can be entirely justified. We also introduce simple examples of semilinear equations that can be treated easily based on the notion of weak solutions.