Entire solutions for the heat equation

Abstract

We consider the solutions of the heat equation ∂tF = ∂2zF which are entire in z and t (caloric functions). 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. Also, regarding the zeros zk(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 ℂ2 and, then, we derive the t-evolution equations of zk(t). These are differential equations that hold for all but countably many ts in ℂ.