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

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Abstract.

We consider the dead-core problem for the semilinear heat equation with strong absorption u t =u xx −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.

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.

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Authors and Affiliations

  1. Department of Mathematics, National Taiwan Normal University, 88, Sec. 4, Ting Chou Road, Taipei, 117, Taiwan

    Jong-Shenq Guo

  2. Département de Mathématiques, INSSET, Université de Picardie, 02109, St-Quentin, France

    Philippe Souplet

  3. Laboratoire de Mathématiques Appliquées, UMR CNRS 7641, Université de Versailles, 45 avenue des Etats-Unis, 78035, Versailles, France

    Philippe Souplet

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  1. Jong-Shenq Guo
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  2. Philippe Souplet
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Correspondence to Philippe Souplet.

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This work was partially supported by the National Science Council of the Republic of China under the grants NSC 91-2735-M-001-001 and NSC 92-2115-M-003-008.

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Guo, JS., Souplet, P. Fast rate of formation of dead-core for the heat equation with strong absorption and applications to fast blow-up. Math. Ann. 331, 651–667 (2005). https://doi.org/10.1007/s00208-004-0601-7

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