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 $A_2\ge A_1= (2α_1(1-\3)(n+2α_1-2))^{-1/(1+ν)}$ where $0<\3<1$ is a constant, the global solution $u$ exists and satisfies $A_1(1+|x{|}^2+b_{1}t{)}^{{\alpha }_1}\le u(x,t)\le A_2(1+|x{|}^2+b_{2}t{)}^{{\alpha }_2}$ in ${\mathbb{R}}^n\times (0,\infty )$ where $b_1=2(n+2α_1-2)\3$ 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.

16 pages

References in corpus (1)

• Global and touchdown behaviour of the generalized MEMS device equation