Analytic Solutions of the Heat Equation

Abstract

Motivated by the recent proof of Newman's conjecture [12] we study certain properties of entire caloric functions, namely solutions of the heat equation ∂tF = ∂ 2 z F which are entire in z and t. As a prerequisite, we establish some general properties of the order and type of an entire function. Then, we start our inquiry on entire caloric functions by determining the necessary and sufficient condition for a function f (z) to be the initial condition of an entire solutions of the heat equation and, subsequently, we examine the relation of the z-order and z-type of an entire caloric function F (t, z), viewed as function of z, to its t-order and t-type respectively, if it is viewed as function of t. After that, we shift our attention to the zeros z k (t) of an entire caloric function F (t, z), viewed as function of z. We show that the points (t, z) at which F (t, z) = ∂zF (t, z) = 0 form a discrete set in C 2 and we derive the t-evolution equations of the zeros of F (t, z). These are differential equations which hold for all but countably many t ∈ C.