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Fast rate of formation of dead-core for the heat equation with strong absorption and applications to fast blow-up. (English) Zbl 1063.35087
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 $a(t,x)$. 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. 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.

35K60Nonlinear initial value problems for linear parabolic equations
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
Full Text: DOI
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