The Cauchy problem for the Finsler heat equation

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 \...

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Veröffentlicht in:	arXiv.org 2017-10
Hauptverfasser:	Akagi, Goro, Ishige, Kazuhiro, Sato, Ryuichi
Format:	Artikel
Sprache:	eng
Schlagworte:	Cauchy problems Laplace transforms Linear operators Nonlinear programming Thermodynamics
Online-Zugang:	Volltext
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Zusammenfassung:	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 $\partial_t:=\partial/\partial t$.
ISSN:	2331-8422

Zusammenfassung:	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 \(\partial_t:=\partial/\partial t\).
ISSN:	2331-8422