Existence and convexity of solutions of the fractional heat equation

Abstract

We prove that the initial-value problem for the fractional heat equation admits an entire solution provided that the (possibly unbounded) initial datum has a conveniently moderate growth at infinity. Under the same growth condition we also prove that the solution is unique. The result does not require any sign assumption, thus complementing the Widder’s type theorem of Barrios et al. [1] for positive solutions. Finally, we show that the fractional heat flow preserves convexity of the initial datum. Incidentally, several properties of stationary convex solutions are established

Fractional thoughts

“…More recently, several authors have analyzed properties such as estimates of the heat kernel, the weak Harnack inequality and the Hölder continuity of solutions, interior and at the boundary, for more general nonlocal parabolic equations, results of Fujita type, a Widder type theorem, Nash-type inequalities, eigenvalue estimates, nonlocal porous medium equation etc., see e.g. [Ko95], [BLW05], [BJ07], [BM07], [BGR10], [CKK11], [DQRV12], [FK13], [BP13], [BPSV14], [AMPP16], [BSV17], [Fr17], [GI17], [V17], [V17'] but this list is by no means exhaustive. The regularity theory for very general nonlocal parabolic operators has been developed in the paper [CCV11].…”

An inverse problem for a fractional diffusion equation with fractional power type nonlinearities

“…It has been shown that the associated kernel K(x, t) is smooth and positive in R n × R + . See [13,25] for more details. The following estimate was given in Lemma 3.1 in [25].…”

Cauchy problem for a fractional anisotropic parabolic equation in anisotropic Hölder spaces

“…Next we mention shortly several results devoted to equations with either fractional space operator or with fractional time derivative. Among the papers, concerning Cauchy problems for equation with the usual first derivative in time and with either fractional Laplacian or with a more general nonlocal space operator we refer to the papers [12], [14], [20], [35], [37], [47]. These papers contain, in particular, results on Schauder estimates and regularity in Hölder or Hölder -like classes.…”