Optimal existence classes and nonlinear-like dynamics in the linear heat equation in Rd

Abstract

We analyse the behaviour of solutions of the linear heat equation in Rd for initial data in the classes Mε (Rd ) of Radon measures with e−ε|x|2 d|u0| < ∞. We show that Rd these classes are optimal for local and global existence of non-negative solutions: in particular M0(Rd) := ∩ε>0Mε(Rd) consists of those initial data for which a solution of the heat equation can be given for all time using the heat kernel representation formula. We prove existence, uniqueness, and regularity results for such initial data, which can grow rapidly at infinity, and then show that they give rise to properties associated more often with nonlinear models. We demonstrate the finite-time blowup of solutions, showing that the set of blowup points is the complement of a convex set, and that given any closed convex set there is an initial condition whose solutions remain bounded precisely on this set at the ‘blowup time’. We also show that wild oscillations are possible from non-negative initial data as t → ∞ and that one can prescribe