Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces

Kazuhiro Ishige; Ryuichi  Sato

doi:10.3934/dcds.2016.36.2627

Article Contents

2016, Volume 36, Issue 5: 2627-2652. Doi: 10.3934/dcds.2016.36.2627

This issue Previous Article Persistence properties and unique continuation for a dispersionless two-component Camassa-Holm system with peakon and weak kink solutions Next Article Invariance properties of the Monge-Kantorovich mass transport problem

Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces

Kazuhiro Ishige^1, and
Ryuichi Sato^1,

1.
Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578

Received: March 31, 2015

Revised: July 31, 2015

Published: May 2017

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Abstract

We establish the local existence and the uniqueness of solutions of the heat equation with a nonlinear boundary condition for the initial data in uniformly local $L^r$ spaces. Furthermore, we study the sharp lower estimates of the blow-up time of the solutions with the initial data $\lambda\psi$ as $\lambda\to 0$ or $\lambda\to\infty$ and the lower blow-up estimates of the solutions.

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Mathematics Subject Classification: Primary: 35B44, 35K55, 35K60.

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