Diffusion versus absorption in semilinear parabolic problems

نویسندگان

  • Andrey Shishkov
  • Laurent Véron
چکیده

We study the limit, when k → ∞, of the solutions u = uk of (E) ∂tu−∆u+ h(t)uq = 0 in RN × (0,∞), uk(., 0) = kδ0, with q > 1, h(t) > 0. If h(t) = e−ω(t)/t where ω > 0 satisfies to R 1 0 p ω(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 ∂tu−∆u + h(t)uq = 0 with m > 1 and ∂tu−∆u+ h(t)eu = 0. Diffusion versus absorption dans des problèmes paraboliques semilinéaires Résumé. Nous étudions la limite, quand k → ∞, des solutions u = uk de (E) ∂tu−∆u+ h(t)uq = 0 dans RN × (0,∞), uk(., 0) = kδ0 avec q > 1, h(t) > 0. Nous montrons que si h(t) = e−ω(t)/t où ω > 0 vérifie R 1 0 p ω(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 ∂tu − ∆um + h(t)uq = 0 avec m > 1 et ∂tu−∆u+ h(t)eu = 0. Version française abrégée Soit q > 1 et h : R+ 7→ R+ une fonction continue, croissante telle que h(t) > 0 pour t > 0. Il est facile de vérifier que toute solution positive u de (1) ∂tu−∆u+ h(t)u = 0 dans R×]0,+∞[ satisfait à (2) u(x, t) ≤ U(t) := (

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تاریخ انتشار 2008