UNBOUNDED SOLUTIONS OF THE NONLOCAL HEAT EQUATION

We consider the Cauchy problem posed in the whole space for the following nonlocal heat equation: u(t) = J * u - u, where J is a symmetric continuous probability density. Depending on the tail of J, we give a rather complete picture of the problem in optimal classes of data by: (i) estimating the initial trace of (possibly unbounded) solutions; (ii) showing existence and uniqueness results in a suitable class; (iii) proving blow-up in finite time in the case of some critical growths; (iv) giving explicit unbounded polynomial solutions.