Results of existence and uniqueness for the Cauchy problem of semilinear heat equations on stratified Lie groups

Abstract

The aim of this paper is to give existence and uniqueness results for solutions of the Cauchy problem for semilinear heat equations on stratified Lie groups $\mathbb{G}$ with the homogeneous dimension $N$. We consider the nonlinear function behaves like $|u|^α$ or $|u|^{α-1}u$ $(α>1)$ and the initial data $u_0$ belongs to the Sobolev spaces $L^p_s(\mathbb{G})$ for $1<p<\infty$ and $0<s<N/p$. Since stratified Lie groups $\mathbb{G}$ include the Euclidean space ${\mathbb R}^n$ as an example, our results are an extension of the existence and uniqueness results obtained by F. Ribaud on ${\mathbb R}^n$ to $\mathbb{G}$. It should be noted that our proof is very different from it given by Ribaud on ${\mathbb R}^n$. We adopt the generalized fractional chain rule on $\mathbb{G}$ to obtain the estimate for the nonlinear term, which is very different from the paracomposition technique adopted by Ribaud on ${\mathbb R}^n$. By using the generalized fractional chain rule on $\mathbb{G}$, we can avoid the discussion of Fourier analysis on $\mathbb{G}$ and make the proof more simple.