Existence and convexity of solutions of the fractional heat equation

We prove that the initial-value problem for the fractional heat equation admits a 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. Our result does not require any sign assumption, thus complementing the Widder's type theorem of Barrios et al. (Arch. Rational Mech. Anal. 213 (2014) 629-650) 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.