Self-similar extinction for a fast diffusion equation with weighted absorption

Abstract

Finite time extinction of any bounded solution to the fast diffusion equation with spatially inhomogeneous absorption $${\partial }_{t}u=\hat{I}”u^m-|x{|}^{\ddot{I}}ƒu^p,(x,t)\in {\mathbb{R}}^N\times (0,\infty ),$$ with $N\ge 1$ and exponents $$p>1,m_c=\frac{(N-2{)}_{+}}{N}<m<1,\ddot{I}ƒ>\ddot{I}{ƒ}_{\ast }:=\frac{2(p-1)}{1-m},$$ is established. Moreover, the existence of self-similar solutions of the form $$U(x,t)=(T-t{)}^{\hat{I}}\pm f(|x|(T-t{)}^2),\hat{I}\pm =\frac{\ddot{I}ƒ+2}{(1-m)(\ddot{I}ƒ-\ddot{I}{ƒ}_{\ast })},{\hat{I}}^2=\frac{p-m}{(1-m)(\ddot{I}ƒ-\ddot{I}{ƒ}_{\ast })},$$ with $f(0)>0$, $f^{\prime }(0)=0$ and $$\underset{\hat{I}¾\to \infty }{\lim }\hat{I}{¾}^{(\ddot{I}ƒ+2)/(p-m)}f(\hat{I}¾)=L\in (0,\infty ).$$ is proved, together with some unbounded self-similar solutions as well. The property of finite time extinction is in striking contrast to the standard fast diffusion equation with absorption (that is, $\ddot{I}ƒ=0$), where the strict positivity of solutions for any $t\in (0,\infty )$ is well-known.