Existence and Uniqueness for the Cauchy Problem of Semilinear Heat Equations on Stratified Lie Groups in the Critical Sobolev Space

Abstract

The aim of this paper is to give existence and uniqueness results for solutions to the Cauchy problem of semilinear heat equations on stratified Lie groups $\mathbb{G}$ with the small initial data belonging to the critical Sobolev space. We consider a power type nonlinearity that behaves like $|u|^{\alpha}$ or $|u|^{\alpha-1}u$ ($\alpha > 1$).