  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 $|u|^{\alpha}$ or $|u|^{\alpha-1}u$
$(\alpha>1)$ 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.
