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

TL;DR

This paper proves local existence and uniqueness of solutions for a heat equation with nonlinear boundary conditions in uniformly local $L^r$ spaces, and analyzes the blow-up time estimates for scaled initial data.

Contribution

It introduces new results on existence, uniqueness, and blow-up time estimates for the heat equation with nonlinear boundary conditions in uniformly local $L^r$ spaces.

Findings

Established local existence and uniqueness of solutions.

Derived sharp lower estimates for blow-up times as initial data scales.

Analyzed lower blow-up estimates for solutions with scaled initial data.

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.