Nonlinear boundary value problems relative to one dimensional heat equation

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.

22 pages, 16 ref. Rendiconti dell'Istituto di Matematica dell'Universit{à} di Trieste: an International Journal of Mathematics, Universit{à} di Trieste, In press