Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1063.35087
Guo, Jong-Shenq; Souplet, Philippe
Fast rate of formation of dead-core for the heat equation with strong absorption and applications to fast blow-up. (English)
[J] Math. Ann. 331, No. 3, 651-667 (2005). ISSN 0025-5831; ISSN 1432-1807/e

Summary: We consider the dead-core problem for the semilinear heat equation with strong absorption $u_t = u_{xx} - u^p$ 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 $-a(t,x)u^p$ for a suitable bounded, uniformly positive function $a(t,x)$. \par 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. \par Sharp estimates are also obtained for the profiles of dead-core and blow-up. The blow up profile turns out to be slightly less singular than in the unperturbed case.

MSC 2000:: *35K60 (Nonlinear) BVP for (non)linear parabolic equations
35B05 General behavior of solutions of PDE

Keywords: positive boundary values; extinction; quenching

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]