# American Institute of Mathematical Sciences

March  2010, 9(2): 397-411. doi: 10.3934/cpaa.2010.9.397

## Fast rate of dead core for fast diffusion equation with strong absorption

 1 College of mathematics and physics, Chongqing University, Chongqing, 400044, China 2 College of mathematics and physics, Chongqing University, Chongqing, 400044, School of mathematics and statistics, Southwest University, Chongqing, 400715, China 3 Department of Mathematics, Sichuan Normal University, Chengdu, 610066, China

Received  January 2009 Revised  July 2009 Published  December 2009

This paper deals with the dead core problem for the fast diffusion equation with strong absorption and positive boundary values. We prove that the dead core rate is faster than the one given by the corresponding ODE, which is contrary to the known results for the related extinction, quenching and blow up problems. Moreover, we find the dead core rate is quite unstable: the ODE rate can be recovered if the absorption term is replaced by $-a(t,x)u^p$ for a suitable bounded, uniformly positive function $a(t,x)$. As an application of the above results, some new and relatively simple examples of fast blow up are provided, and a phenomenon of strong sensitivity to gradient perturbations is exhibited. Furthermore, the blow up rate is found to depend on a constant in the perturbation term, and sharp estimates are also obtained for the profile of dead core and blow up.
Citation: Chunlai Mu, Jun Zhou, Yuhuan Li. Fast rate of dead core for fast diffusion equation with strong absorption. Communications on Pure & Applied Analysis, 2010, 9 (2) : 397-411. doi: 10.3934/cpaa.2010.9.397
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