Fractional semi-linear parabolic equations with unbounded data

Alibaud, Nathaël; Imbert, Cyril

doi:10.1090/S0002-9947-08-04758-2

Fractional semi-linear parabolic equations with unbounded data
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by Nathaël Alibaud and Cyril Imbert PDF

Trans. Amer. Math. Soc. 361 (2009), 2527-2566 Request permission

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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