Uniqueness/nonuniqueness for nonnegative solutions of the Cauchy problem for $u_t=Δu-u^p$ in a punctured space

Abstract

Consider classical solutions to the following Cauchy problem in a punctured space: $ &u_t=Δu -u^p \text{in} (R^n-\{0\})\times(0,\infty); & u(x,0)=g(x)\ge0 \text{in} R^n-\{0\}; &u\ge0 \text{in} (R^n-\{0\})\times[0,\infty). $ We prove that if $p\ge\frac n{n-2}$, then the solution to \eqref{abstract} is unique for each $g$. On the other hand, if $p<\frac n{n-2}$, then uniqueness does not hold when $g=0$; that is, there exists a nontrivial solution with vanishing initial data.