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Growth rate and extinction rate of a reaction diffusion equation with a singular nonlinearity. (English) Zbl 1240.35222

Summary: 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 \(A_1(1+| x| ^2)^{\alpha _1}\leq u_{0}\leq A_2(1+| x| ^2)^{\alpha _2}\) in \(\mathbb {R}^n\) for some constants \(1/(1+\nu )\leq \alpha _1<1,\, \alpha _2\geq \alpha _1\) and \(A_2\geq A_1=(2\alpha _1(1-\epsilon )(n+2\alpha _1-2))^{-1/(1+\nu )}\), where \(0<\epsilon <1\) is a constant, the global solution \(u\) exists and satisfies \(A_1(1+| x| ^2+b_1t)^{\alpha _1}\leq u(x,t)\leq A_2(1+| x| ^2+b_2t)^{\alpha _2}\) in \(\mathbb {R}^n\times (0,\infty )\), where \(b_1=2(n+2\alpha _1-2)\epsilon \) and \(b_2=2n\) if \(0<a_2\leq 1\) and \(b_2=2(n+2\alpha _2-2)\) if \(\alpha _2\geq 1\). When \(0<u_{0}\leq A(T_1+| x| ^2)^{1/(1+\nu )}\) in \(\mathbb {R}^{n}\) for some constant \(0<A <((1+\nu )/2n)^{1/(1+\nu )}\), we prove that \(u(x,t)\leq A(b(T-t)+| x| ^2)^{1/(1+\nu )}\) in \(\mathbb {R}^{n}\times (0,\infty )\) 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.

MSC:

35K15 Initial value problems for second-order parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K57 Reaction-diffusion equations
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