The Cauchy problem for the Finsler heat equation

Abstract

Abstract Let H be a norm of ℝ N {\mathbb{R}^{N}} and H 0 {H_{0}} the dual norm of H. Denote by Δ H {\Delta_{H}} the Finsler–Laplace operator defined by Δ H ⁢ u := div ⁡ ( H ⁢ ( ∇ ⁡ u ) ⁢ ∇ ξ ⁡ H ⁢ ( ∇ ⁡ u ) ) {\Delta_{H}u:=\operatorname{div}(H(\nabla u)\nabla_{\xi}H(\nabla u))} . In this paper we prove that the Finsler–Laplace operator Δ H {\Delta_{H}} acts as a linear operator to H 0 {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 ∂ t ⁡ u = Δ H ⁢ u , x ∈ ℝ N , t > 0 , \partial_{t}u=\Delta_{H}u,\quad x\in\mathbb{R}^{N},\,t>0, where N ≥ 1 {N\geq 1} and ∂ t := ∂ ∂ ⁡ t {\partial_{t}:=\frac{\partial}{\partial t}} .