Fast rate of formation of dead-core for the heat equation with strong absorption and applications to fast blow-up
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- 作者：Jong-Shenq Guo and Philippe Souplet
- 刊名：Mathematische Annalen
- 出版年：2005
- 出版时间：March 2005
- 年：2005
- 卷：331
- 期：3
- 页码：651-667
- 全文大小：178 KB
- 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Mathematics - 出版者：Springer Berlin / Heidelberg
- ISSN：1432-1807
文摘
We consider the dead-core problem for the semilinear heat equation with strong absorption ut=uxx–up with 0<p<1 and positive boundary values. We investigate the dead-core rate, i.e. the rate at which the solution reaches its first zero. Surprisingly, we find that the dead-core rate is faster than the one given by the corresponding ODE. This stands in sharp contrast with known results for the related extinction, quenching and blow-up problems. Moreover, we find that the dead-core rate is actually quite unstable: the ODE rate can be recovered if the absorption term is replaced by –a(t,x)up for a suitable bounded, uniformly positive function a(t,x).The result has some unexpected consequences for blow-up problems with perturbations. Namely, we obtain the conclusion that perturbing the standard semilinear heat equation by a dissipative gradient term may lead to fast blow-up, a phenomenon up to now observed only in supercritical higher dimensional cases for the unperturbed problem. Furthermore, the blow-up rate is found to depend on a very sensitive way on the constant in factor of the perturbation term.