Fast rate of formation of dead-core for the heat equation with strong absorption and applications to fast blow-up

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.