Growth rate and extinction rate of a reaction diffusion equation with a singular nonlinearity

Abstract

We prove the growth rate of global solutions of the equation $u_t=\Delta u-u^{-\nu}$ in $\R^n\times (0,\infty)$, $u(x,0)=u_0>0$ in $\R^n$, where $\nu>0$ is a constant. More precisely for any $0<u_0\in C(\R^n)$ satisfying $A_1(1+|x|^2)^{\alpha_1}\le u_0\le A_2(1+|x|^2)^{\alpha_2}$ in $\R^n$ for some constants $1/(1+\nu)\le\alpha_1<1$, $\alpha_2\ge\alpha_1$ and $A_2\ge A_1= (2\alpha_1(1-\3)(n+2\alpha_1-2))^{-1/(1+\nu)}$ where $0<\3<1$ is a constant, the global solution $u$ exists and satisfies $A_1(1+|x|^2+b_1t)^{\alpha_1}\le u(x,t)\le A_2(1+|x|^2+b_2t)^{\alpha_2}$ in $\R^n\times (0,\infty)$ where $b_1=2(n+2\alpha_1-2)\3$ and $b_2=2n$ if $0<\alpha_2\le 1$ and $b_2=2(n+2\alpha_2-2)$ if $\alpha_2>1$. We also find various conditions on the initial value for the solution to extinct in a finite time and obtain the corresponding decay rate of the solution near the extinction time.