On Positive Solutions of the Heat Equation

Nagoya Mathematical Journal ◽

10.1017/s0027763000012472 ◽

1967 ◽

Vol 30 ◽

pp. 203-207 ◽

Cited By ~ 2

Author(s):

Masasumi Kato

Keyword(s):

Cauchy Problem ◽

Euclidean Space ◽

Heat Equation ◽

Uniqueness Theorem ◽

Positive Solutions ◽

Dimensional Euclidean Space ◽

The Cauchy Problem ◽

The Uniqueness Theorem

Consider the positive and twice continuously differentiable solutions u of the heat equationin an open t-strip Ω = Rn×(0,T) for some T>0, where Rn is the n-dimensional Euclidean space.In this note, we prove a theorem of Fatou type on u and, as its application, the uniqueness theorem for the Cauchy problem of ( 1 ).

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An Application Of The Theory Of Scale Of Banach Spaces

Annales Mathematicae Silesianae ◽

10.1515/amsil-2015-0005 ◽

2015 ◽

Vol 29 (1) ◽

pp. 51-59

Author(s):

Łukasz Dawidowski

Keyword(s):

Banach Space ◽

Cauchy Problem ◽

Boundary Conditions ◽

Heat Equation ◽

Banach Spaces ◽

Abstract Cauchy Problem ◽

Initial Boundary ◽

Scale Of Banach Spaces ◽

The Cauchy Problem ◽

Selection Of

AbstractThe abstract Cauchy problem on scales of Banach space was considered by many authors. The goal of this paper is to show that the choice of the space on scale is significant. We prove a theorem that the selection of the spaces in which the Cauchy problem ut − Δu = u|u|s with initial–boundary conditions is considered has an influence on the selection of index s. For the Cauchy problem connected with the heat equation we will study how the change of the base space influents the regularity of the solutions.

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The Cauchy problem for the Finsler heat equation

Advances in Calculus of Variations ◽

10.1515/acv-2017-0048 ◽

2020 ◽

Vol 13 (3) ◽

pp. 257-278 ◽

Cited By ~ 1

Author(s):

Goro Akagi ◽

Kazuhiro Ishige ◽

Ryuichi Sato

Keyword(s):

Cauchy Problem ◽

Linear Operator ◽

Heat Equation ◽

Laplace Operator ◽

Sufficient Condition ◽

Smooth Functions ◽

Dual Norm ◽

The Cauchy Problem

AbstractLet H be a norm of {\mathbb{R}^{N}} and {H_{0}} the dual norm of H. Denote by {\Delta_{H}} the Finsler–Laplace operator defined by {\Delta_{H}u:=\operatorname{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,\quad x\in\mathbb{R}^{N},\,t>0,where {N\geq 1} and {\partial_{t}:=\frac{\partial}{\partial t}}.

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