Nonlinear boundary value problems relative to one dimensional heat equation

TL;DR

This paper investigates the existence and properties of solutions to a nonlinear boundary value problem related to the one-dimensional heat equation, including self-similar solutions and extensions to higher dimensions.

Contribution

It introduces new existence results for solutions with measure boundary data and analyzes self-similar solutions for nonlinear boundary conditions, extending to higher dimensions.

Findings

Existence of solutions with measure boundary data.

Characterization of self-similar solutions for nonlinear boundary conditions.

Extensions to higher-dimensional problems.

Abstract

We consider the problem of existence of a solution $u$ to $\partial_t u-\partial_{xx} u = 0$ in $(0,T)\times\mathbb{R}_+$ subject to the boundary condition $-u_x(t,0)+g(u(t,0))=\mu$ on $(0,T)$ where $\mu$ is a measure on $(0,T)$ and $g$ a continuous nondecreasing function. When $p>1$ we study the set of self-similar solutions of $\partial_t u-\partial_{xx} u = 0$ in $\mathbb{R}_+\times\mathbb{R}_+$ such that $-u_x(t,0)+u^p=0$ on $(0,\infty)$. At end, we present various extensions to a higher dimensional framework.