Fast rate of dead core for fast diffusion equation with strong absorption
Chunlai Mu, Jun Xing Zhou, Yu-Huan Li · Communications on Pure & Applied Analysis · 2010
This paper deals with the dead core problem for the fast diffusionequation with strong absorption and positive boundary values. Weprove that the dead core rate is faster than the one given by thecorresponding ODE, which is contrary to the known results for therelated extinction, quenching and blow up problems. Moreover, wefind the dead core rate is quite unstable: the ODE rate can berecovered if the absorption term is replaced by $-a(t,x)u^p$ for asuitable bounded, uniformly positive function $a(t,x)$. As anapplication of the above results, some new and relatively simpleexamples of fast blow up are provided, and a phenomenon of strongsensitivity to gradient perturbations is exhibited. Furthermore, theblow up rate is found to depend on a constant in the perturbationterm, and sharp estimates are also obtained for the profile of deadcore and blow up.