A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions

Ahmad El Soufi (1) , Mustapha Jazar (2) , Régis Monneau (3)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 Mathematics department
3 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique

In this paper we study a simple non-local semilinear parabolic equation with Neumann boundary condition. We give local existence result and prove global existence for small initial data. A natural non increasing in time energy is associated to this equation. We prove that the solution blows up at finite time $T$ if and only if its energy is negative at some time before $T$. The proof of this result is based on a Gamma-convergence technique.

Mots clés: Semi-linear parabolic equations; Blow-up; Global existence; Asymptotic behavior of solutions; Gamma convergence; Neumann Heat kernel; Non local term; Comparison principle

Domaines: Equations aux dérivées partielles [math.AP]

Ahmad El Soufi, Mustapha Jazar, Régis Monneau. A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2007, 24, pp.17--39. ⟨hal-00126955⟩