Fractional semi-linear parabolic equations with unbounded data

Nathaël Alibaud (1, 2) , Cyril Imbert (1, 3)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

This paper is devoted to the study of semi-linear parabolic equations whose principal term is fractional, i.e. is integral and eventually singular. A typical example is the fractional Laplace operator. This work sheds light on the fact that, if the initial datum is not bounded, assumptions on the non-linearity are closely related to its behavior at infinity. The sub-linear and super-linear cases are first treated by classical techniques. We next present a third original case: if the associated first order Hamilton-Jacobi equation is such that perturbations propagate at finite speed, then the semi-linear parabolic equation somehow keeps memory of this property. By using such a result, locally bounded initial data that are merely integrable at infinity can be handled. Next, regularity of the solution is proved. Eventually, strong convergence of gradients as the fractional term disappears is proved for strictly convex non-linearity.

Mots clés: integro-differential Hamilton-Jacobi equation; unbounded data; unbounded solutions; finite-infinite propagation speed; non-local regularization; Lévy operator; viscosity solution; convergence of the gradients

Domaines: Equations aux dérivées partielles [math.AP]

Nathaël Alibaud, Cyril Imbert. Fractional semi-linear parabolic equations with unbounded data. Transactions of the American Mathematical Society, 2009, 361, pp.2527-2566. ⟨10.1090/S0002-9947-08-04758-2⟩. ⟨hal-00144548⟩