We prove the growth rate of global solutions of the equation $u_t=Delta u-u^{- u}$ in $R^ntimes (0,infty)$, $u(x,0)=u_0>0$ in $R^n$, where $ u>0$ is a constant. More precisely for any $0<u_0in C(R^n)$ satisfying $A_1(1+|x|^2)^{alpha_1}le u_0le A_2(1+|x|^2)^{alpha_2}$ in $R^n$ for some constants $1/(1+ u)lealpha_1<1$, $alpha_2gealpha_1$ and $A_2ge A_1= (2alpha_1(1-3)(n+2alpha_1-2))^{-1/(1+ u)}$ 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 $R^ntimes (0,infty)$ where $b_1=2(n+2alpha_1-2)3$ and $b_2=2n$ if $0<alpha_2le 1$ and $b_2=2(n+2alpha_2-2)$ 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.
Download