Growth rate and extinction rate of a reaction diffusion equation with a singular nonlinearity
Abstract
Description
We prove the growth rate of global solutions of the equation $u_t=Δu-u^{-ν}$ in $\R^n\times (0,\infty)$, $u(x,0)=u_0>0$ in $\R^n$, where $ν>0$ is a constant. More precisely for any $0<u_0\in C(\R^n)$ satisfying $A_1(1+|x|^2)^{α_1}\le u_0\le A_2(1+|x|^2)^{α_2}$ in $\R^n$ for some constants $1/(1+ν)\leα_1<1$, $α_2\geα_1$ and $A_2\ge A_1= (2α_1(1-\3)(n+2α_1-2))^{-1/(1+ν)}$ where $0<\3<1$ is a constant, the global solution $u$ exists and satisfies $A_1(1+|x|^2+b_1t)^{α_1}\le u(x,t)\le A_2(1+|x|^2+b_2t)^{α_2}$ in $\R^n\times (0,\infty)$ where $b_1=2(n+2α_1-2)\3$ and $b_2=2n$ if $0<α_2\le 1$ and $b_2=2(n+2α_2-2)$ if $α_2>1$. 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.
16 pages
16 pages