The Cauchy problem for the Finsler heat equation

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) ⁢ ∇ ζ ⁡ 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}}.