The Cauchy problem for the Finsler heat equation

TL;DR

This paper investigates the Finsler heat equation, establishing the linearity of the Finsler-Laplace operator on symmetric functions and providing optimal conditions for the existence of solutions to the Cauchy problem.

Contribution

It proves the linearity of the Finsler-Laplace operator on symmetric functions and derives optimal existence conditions for the Finsler heat equation's solutions.

Findings

Finsler-Laplace operator acts linearly on symmetric functions

Derived optimal existence conditions for solutions

Characterized behavior of solutions to the Finsler heat equation

Abstract

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 $\Delta_Hu:=\mbox{div}\,(H(\nabla u)\nabla_\xi H(\nabla u))$. 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 $$ \partial_t u=\Delta_H u,\qquad x\in{\bf R}^N,\quad t>0, $$ where $N\ge 1$ and $\partial_t:=\partial/\partial t$.