A multi-dimensional blow-up problem due to a concentrated nonlinear source in $\mathbb {R}^{N}$

Let $B$ be an $N$-dimensional ball $\left \{ x\in \mathbb {R}^{N}:\left \vert x\right \vert <R\right \}$ centered at the origin with a radius $R$, and $\partial B$ be its boundary. Also, let $\nu \left ( x\right )$ denote the unit inward normal at $x\in \partial B$, and let $\chi _{B}\left ( x\right )$ be the characteristic function, which is 1 for $x\in B$, and $0$ for $x\in \mathbb {R}^{N}\setminus B$. This article studies the following multi-dimensional semilinear parabolic problem with a concentrated nonlinear source on $\partial B$: \[ \begin {array}[c]{c} u_{t}-\triangle u=\alpha {\displaystyle {\frac {\partial \chi _{B}\left ( x\right ) }{\partial \nu }}}f\left ( u\right ) \text { in }\mathbb {R}^{N}\times (0,T],\\ u\left ( x,0\right ) =\psi \left ( x\right ) \text {\ for }x\in \mathbb {R}^{N}\text {, }u\left ( x,t\right ) \rightarrow 0\ \text {as\ }\left \vert x\right \vert \rightarrow \infty \text { for }0<t\leq T, \end {array} \] where $\alpha$ and $T$ are positive numbers, $f$ and $\psi$ are given functions such that $f\left ( 0\right ) \geq 0$, $f\left ( u\right )$ and $f^{\prime }\left ( u\right )$ are positive for $u>0$, $f^{\prime \prime }\left ( u\right ) \geq 0$ for $u>0$, and $\psi$ is nontrivial on $\partial B,$ nonnegative, and continuous such that $\psi \rightarrow 0$ as $\left \vert x\right \vert \rightarrow \infty ,$ $\int _{\mathbb {R}^{N}}\psi \left ( x\right ) dx<\infty$, and $\triangle \psi +\alpha \left ( \partial \chi _{B}\left ( x\right ) /\partial \nu \right ) f\left ( \psi \left ( x\right ) \right ) \geq 0$ in $\mathbb {R}^{N}.$ It is shown that the problem has a unique nonnegative continuous solution before blowup occurs. We assume that $\psi \left ( x\right ) =M\left ( 0\right ) >\psi \left ( y\right )$ for $x\in \partial B$ and $y\notin \partial B$, where $M\left ( t\right ) =\sup _{x\in \mathbb {R}^{N}}u\left ( x,t\right )$. It is proved that if $u$ blows up in a finite time, then it blows up everywhere on $\partial B$. If, in addition, $\psi$ is radially symmetric about the origin, then we show that if $u$ blows up, then it blows up on $\partial B$ only. Furthermore, if $f\left ( u\right ) \geq \kappa u^{p}$, where $\kappa$ and $p$ are positive constants such that $p>1$, then it is proved that for any $\alpha$, $u$ always blows up in a finite time for $N\leq 2$; for $N\geq 3$, it is shown that there exists a unique number $\alpha ^{\ast }$ such that $u$ exists globally for $\alpha \leq \alpha ^{\ast }$ and blows up in a finite time for $\alpha >\alpha ^{\ast }$. A formula for computing $\alpha ^{\ast }$ is given.