The Cauchy problem for the Finsler heat equation

Let $H$ be a norm of ${\mathbf{R}}^N$ and $H_0$ the dual norm of $H$. Denote by ${\mathrm{\Delta }}_H$ the Finsler-Laplace operator defined by ${\mathrm{\Delta }}_{H}u:=div(H(\mathrm{\nabla }u){\mathrm{\nabla }}_{\xi }H(\mathrm{\nabla }u))$. In this paper we prove that the Finsler-Laplace operator ${\mathrm{\Delta }}_H$ 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 $${\mathrm{\partial }}_{t}u={\mathrm{\Delta }}_{H}u,x\in {\mathbf{R}}^N,t>0,$$ where $N\ge 1$ and ${\mathrm{\partial }}_t:=\mathrm{\partial }\mathrm{/}\mathrm{\partial }t$.