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 ${{\mathbb{R}}}^n\times (0,\infty)$, $u(x,0)=u_0>0$ in ${{\mathbb{R}}}^n$, where $\nu>0$ is a constant. More precisely for any $0 <u_0\in C({{\mathbb{R}}}^n)$ satisfying $A_1(1+|x|^2)^{\alpha_1}\le u_0\le A_2(1+|x|^2)^{\alpha_2}$ in ${{\mathbb{R}}}^n$ for some constants $1/(1+\nu)\le\alpha_1 <1$, $\alpha_2\ge\alpha_1$ and $A_2\ge A_1= (2\alpha_1(1-{\varepsilon})(n+2\alpha_1-2))^{-1/(1+\nu)}$ where $0 <{\varepsilon} <1$ 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 ${{\mathbb{R}}}^n\times (0,\infty)$ where $b_1=2(n+2\alpha_1-2){\varepsilon}$ and $b_2=2n$ if $0 <\alpha_2\le 1$ and $b_2=2(n+2\alpha_2-2)$ if $\alpha_2>1$. When $0 <u_0\le A(T_1+|x|^2)^{1/(1+\nu)}$ in ${{\mathbb{R}}}^n$ for some constant $0 <A <((1+\nu)/2n)^{1/(1+\nu)}$, we prove that $u(x,t) \le A(b(T-t)+|x|^2)^{1/(1+\nu)}$ in ${{\mathbb{R}}}^n\times (0,T)$ for some constants $b>0$ and $T>0$. Hence, the solution becomes extinct at the origin at time $T$. We also find various other conditions for the solution to become extinct in a finite time and obtain the corresponding decay rate of the solution near the extinction time.