TY - JOUR

T1 - The Cauchy problem for the Finsler heat equation

AU - Akagi, Goro

AU - Ishige, Kazuhiro

AU - Sato, Ryuichi

N1 - Funding Information:
Goro Akagi was partially supported by the Grant-in-Aid for Scientific Research (B) (No. 16H03946)
Publisher Copyright:
© 2020 Walter de Gruyter GmbH, Berlin/Boston 2020.

PY - 2020/7/1

Y1 - 2020/7/1

N2 - 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}}.

AB - 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}}.

KW - Cauchy problem

KW - Finsler heat equation

KW - linearity

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U2 - 10.1515/acv-2017-0048

DO - 10.1515/acv-2017-0048

M3 - Article

AN - SCOPUS:85043758584

VL - 13

SP - 257

EP - 278

JO - Advances in Calculus of Variations

JF - Advances in Calculus of Variations

SN - 1864-8258

IS - 3

ER -