The Cauchy problem for the Finsler heat equation

Abstract

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

References in corpus (1)

• Gradient bounds and rigidity results for singular, degenerate, anisotropic partial differential equations