The Cauchy problem for the Finsler heat equation

Let $H$ be a norm of ${bf R}^N$ and $H_0$ the dual norm of $H$. Denote by $Delta_H$ the Finsler-Laplace operator defined by $Delta_Hu:=mbox{div},(H( abla u) abla_xi H( abla u))$. In this paper we prove that the Finsler-Laplace operator $Delta_H$ acts as a linear operator to $H_0$-radially symmetric smooth functions. Furthermore, we obtain an optimal sufficient condition for the existence of the solution to the Cauchy problem for the Finsler heat equation $$ partial_t u=Delta_H u,qquad xin{bf R}^N,quad t>0, $$ where $Nge 1$ and $partial_t:=partial/partial t$.