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Fractional semi-linear parabolic equations with unbounded data

Authors: Nathaël Alibaud and Cyril Imbert
Journal: Trans. Amer. Math. Soc. 361 (2009), 2527-2566
MSC (2000): Primary 35B65, 35D05, 35B05, 35K65, 35S30
Published electronically: December 17, 2008
MathSciNet review: 2471928
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Abstract: 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 behaviour at infinity. The sublinear and superlinear 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.

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Additional Information

Nathaël Alibaud
Affiliation: UFR Sciences et techniques, Université de Franche-Comté, 16 route de Gray, 25030 Besançon cedex, France

Cyril Imbert
Affiliation: Centre De Recherche en Mathématiques de la Décision, Université Paris-Dauphine, Pl. de Lattre de Tassigny, 75775 Paris cedex 16, France

Keywords: Semi-linear equation, L\'evy operator, fractional Laplacian, unbounded data, unbounded solutions, viscosity solution, finite-infinite propagation speed, regularity, non-local vanishing viscosity method, convergence of gradients.
Received by editor(s): May 3, 2007
Published electronically: December 17, 2008
Article copyright: © Copyright 2008 American Mathematical Society

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