Asymptotic limit of linear parabolic equations with spatio-temporal degenerated potentials

Pablo Àlvarez-Caudevilla (1) , Matthieu Bonnivard (2) , Antoine Lemenant (2)

1 Université Carlos III de Madrid
2 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions

In this paper, we observe how the heat equation in a noncylindrical domain can arise as the asymptotic limit of a parabolic problem in a cylindrical domain, by adding a potential that vanishes outside the limit domain. This can be seen as a parabolic version of a previous work by the first and last authors, concerning the stationary case [Alvarez-Caudevilla and Lemenant, Adv. Differ. Equ. 15 (2010) 649-688]. We provide a strong convergence result for the solution by use of energetic methods and Γ-convergence technics. Then, we establish an exponential decay estimate coming from an adaptation of an argument due to B. Simon.

Mots clés: Parabolic problems; Gamma-convergence; energetic methods; variational methods; partial differential equations

Domaines: Mathématiques [math]

Pablo Àlvarez-Caudevilla, Matthieu Bonnivard, Antoine Lemenant. Asymptotic limit of linear parabolic equations with spatio-temporal degenerated potentials. ESAIM: Control, Optimisation and Calculus of Variations, 2020, 26, pp.50. ⟨10.1051/cocv/2019023⟩. ⟨hal-02929871⟩