Sharp estimate of the life span of solutions to the heat equation with a nonlinear boundary condition

Abstract

Consider the heat equation with a nonlinear boundary condition $${\partial }_{t}u=\Delta u,x\in {\mathbf{R}}_{+}^N,t>0,{\partial }_{\nu }u=u^p,x\in \partial {\mathbf{R}}_{+}^N,t>0,u(x,0)=\kappa \psi (x),x\in D:=\overline{{\mathbf{R}}_{+}^N},$$ where $N\ge 1$, $p>1$, $\kappa >0$ and $\psi$ is a nonnegative measurable function in ${\mathbf{R}}_{+}^N:=\{y\in {\mathbf{R}}^N:y_N>0\}$. Let us denote by $T(\kappa \psi )$ the life span of solutions to this problem. We investigate the relationship between the singularity of $\psi$ at the origin and $T(\kappa \psi )$ for sufficiently large $\kappa >0$ and the relationship between the behavior of $\psi$ at the space infinity and $T(\kappa \psi )$ for sufficiently small $\kappa >0$. Moreover, we give an optimal estimate to $T(\kappa \psi )$, as $\kappa \to \infty$ or $\kappa \to +0$.