Partial Differential Equations
Diffusion versus absorption in semilinear parabolic problems
Comptes Rendus. Mathématique, Volume 342 (2006) no. 8, pp. 569-574.

We study the limit, when k, of the solutions u=uk of (E) tuΔu+h(t)uq=0 in RN×(0,), uk(,0)=kδ0, with q>1, h(t)>0. If h(t)=eω(t)/t where ω>0 satisfies to 01ω(t)t−1dt<, the limit function u is a solution of (E) with a single singularity at (0,0), while if ω(t)1, u is the maximal solution of (E). We examine similar questions for equations such as tuΔum+h(t)uq=0 with m>1 and tuΔu+h(t)eu=0.

Nous étudions la limite, quand k, des solutions u=uk de (E) tuΔu+h(t)uq=0 dans RN×(0,), uk(,0)=kδ0 avec q>1, h(t)>0. Nous montrons que si h(t)=eω(t)/tω>0 vérifie 01ω(t)t−1dt<, la fonction limite u est une solution of (E) avec une singularité isolée en (0,0), alors que si ω(t)1, u est la solution maximale de (E). Nous examinons des questions semblables pour des équations des type suivants tuΔum+h(t)uq=0 avec m>1 et tuΔu+h(t)eu=0.

Accepted:
Published online:
DOI: 10.1016/j.crma.2006.01.021
Shishkov, Andrey 1; Véron, Laurent 2

1 Institute of Applied Mathematics and Mechanics of NAS of Ukraine, R. Luxemburg str. 74, 83114 Donetsk, Ukraine
2 Laboratoire de mathématiques et physique théorique, CNRS UMR 6083, faculté des sciences, 37200 Tours, France
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Shishkov, Andrey; Véron, Laurent. Diffusion versus absorption in semilinear parabolic problems. Comptes Rendus. Mathématique, Volume 342 (2006) no. 8, pp. 569-574. doi : 10.1016/j.crma.2006.01.021. http://www.numdam.org/articles/10.1016/j.crma.2006.01.021/

[1] Brezis, H.; Peletier, L.A.; Terman, D. A very singular solution of the heat equation with absorption, Arch. Rational Mech. Anal., Volume 96 (1985), pp. 185-209

[2] Galaktionov, V.A.; Shishkov, A.E. Saint-Venant's principle in blow-up for higher-order quasilinear parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, Volume 133 (2003), pp. 1075-1119

[3] Marcus, M.; Véron, L. Initial trace of positve solutions to semilinear parabolic inequalities, Adv. Nonlinear Stud., Volume 2 (2002), pp. 395-436

[4] Oleinik, O.A.; Radkevich, E.V. Method of introducing of a parameter in evolution equation, Russian Math. Surveys, Volume 33 (1978), pp. 7-74

[5] Peletier, L.A.; Terman, D. A very singular solution of the porous media equation with absorption, J. Differential Equations, Volume 65 (1986), pp. 396-410

[6] Shishkov, A.E. Dead cores and instantaneous compactification of the supports of energy solutions of quasilinear parabolic equations of arbitrary order, Sbornik: Mathematics, Volume 190 (1999) no. 12, pp. 1843-1869

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