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- W2962958711 abstract "We prove the growth rate of global solutions of the equation $u_t=Delta u-u^{-nu}$ in ${{mathbb{R}}}^ntimes (0,infty)$, $u(x,0)=u_0>0$ in ${{mathbb{R}}}^n$, where $nu>0$ is a constant. More precisely for any $0 <u_0in C({{mathbb{R}}}^n)$ satisfying $A_1(1+|x|^2)^{alpha_1}le u_0le A_2(1+|x|^2)^{alpha_2}$ in ${{mathbb{R}}}^n$ for some constants $1/(1+nu)lealpha_1 <1$, $alpha_2gealpha_1$ and $A_2ge A_1= (2alpha_1(1-{varepsilon})(n+2alpha_1-2))^{-1/(1+nu)}$ 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 ${{mathbb{R}}}^ntimes (0,infty)$ where $b_1=2(n+2alpha_1-2){varepsilon}$ and $b_2=2n$ if $0 <alpha_2le 1$ and $b_2=2(n+2alpha_2-2)$ if $alpha_2>1$. When $0 <u_0le A(T_1+|x|^2)^{1/(1+nu)}$ 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 ${{mathbb{R}}}^ntimes (0,T)$ 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." @default.
- W2962958711 created "2019-07-30" @default.
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- W2962958711 date "2009-07-01" @default.
- W2962958711 modified "2023-10-18" @default.
- W2962958711 title "Growth rate and extinction rate of a reaction diffusion equation with a singular nonlinearity" @default.
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- W2962958711 doi "https://doi.org/10.57262/die/1356019547" @default.
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