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

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  • 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.
    Mathematics Subject Classification: Primary: 35K20, 35K55; Secondary: 35K57.


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