An Exterior Parabolic Differential Inequality Under Semilinear Dynamical Boundary Conditions

Abstract

We study the existence and nonexistence of global weak solutions to the semilinear parabolic differential inequality

$$\begin{aligned} \partial _t u-\Delta u \ge |u|^p,\quad (t,x)\in (0,\infty )\times B^c, \end{aligned}$$

where $p>1$, B is the closed unit ball in ${\mathbb {R}}^N$ ($N\ge 2$) and $B^c$ is its complement, under the semilinear dynamical boundary conditions

$$\begin{aligned} \partial _t u+u \ge |u|^q +w(x), \quad (t,x)\in (0,\infty )\times \partial B \end{aligned}$$

or

$$\begin{aligned} \partial _t u+\partial _\nu u +\alpha u \ge |u|^q +w(x), \quad (t,x)\in (0,\infty )\times \partial B, \end{aligned}$$

where $q>1$, $\alpha \ge 0$, $\partial _\nu :=\frac{\partial }{\partial \nu ^+}$, $\nu ^+$ is the outward unit normal (relative to $B^c$) on $\partial B$ and $w\in L^1(\partial B)$, $\int _{\partial B} w(x)\,\hbox {d}S_x\ge 0$. The cases $\int _{\partial B} w(x)\,\hbox {d}S_x= 0$ and $\int _{\partial B} w(x)\,\hbox {d}S_x>0$ are discussed separately.