Optimal existence and uniqueness theory for the fractional heat equation

We construct a theory of existence, uniqueness and regularity of solutions for the fractional heat equation partial derivative(t)u + (-Delta)(s)u = 0, 0 < s < 1, posed in the whole space RN with data in a class of locally bounded Radon measures that are allowed to grow at infinity with an optimal growth rate. We consider a class of nonnegative weak solutions and prove that there is an equivalence between nonnegative data and solutions, which is given in one direction by the representation formula, in the other one by the initial trace. We review many of the typical properties of the solutions, in particular we prove optimal pointwise estimates and new Harnack inequalities. (C) 2016 Elsevier Ltd. All rights reserved.