Local Well-posedness for Semilinear Heat Equations on H type Groups

Abstract

In this paper, we discuss the local existence and uniqueness for the Cauchy problem of semi heat equations with an initial data in the space $L^q$ on H type group $\mathbb{H}_{p}^{d}$, which has the dimension $p$ of the center, like the argument on the Euclidean space given by F. B. Weissler. That is, the Cauchy problem \[ \begin{cases} \left( \partial_t - \Delta_{\mathbb{H}_{p}^{d}} \right) u(g,t) = |u|^{r-1} u, \quad g \in \mathbb{H}_{p}^{d}, \; t \gt 0, \\ u(g,0) = u_0(g) \in L^q(\mathbb{H}_{p}^{d}) \end{cases} \] has a unique solution if $q \gt N(r-1)/2$ ($q = N(r-1)/2$) and $q \geq r$ ($q \gt r$), where $r \gt 1$ and $N = 2d+2p$ is the homogeneous dimension of $\mathbb{H}_{p}^{d}$.