Solvability of the Heat Equation with a Nonlinear Boundary Condition

Abstract

In this paper we obtain necessary conditions and sufficient conditions for the solvability of the problem $({\rm P})\, \{ \partial_t u=\Delta u,\, x\in{\bf R}^N_+,\,t>0; \partial_\nu u=u^p,\, x\in\partial{\bf R}^N_+,\, t>0; u(x,0)=\mu(x)\ge 0,\, x\in D:=\overline{{\bf R}^N_+} \}$, where $N\ge 1$, $p>1$, and $\mu$ is a nonnegative measurable function in ${\bf R}^N_+$ or a Radon measure in ${\bf R}^N$ with $\mbox{supp}\,\mu\subset D$. Our sufficient conditions and necessary conditions enable us to identify the strongest singularity of the initial data for the solvability for problem $({\rm P})$.