Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 44, pp. 1-25.

Title: Entire solutions for the heat equation
       
Authors: Vassilis G. Papanicolaou, Eva Kallitsi, George Smyrlis

Abstract: 
 We consider the solutions of the heat equation
 $$
 \partial_t F = \partial_z^2 F
 $$
 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 z<sub>k</sub>(t) of an entire caloric function F(t, z), viewed as
 function of z, we show that the points (t, z) at which
 $$
 F(t, z) = \partial_z F(t, z) = 0
 $$
 form a discrete set in $\mathbb{C}^2$ and, then, we derive the t-evolution equations
 of z<sub>k</sub>(t). These are differential equations that hold for all but countably many
 ts in $\mathbb{C}$.

Submitted October 20, 2020. Published May 25, 2021.
Math Subject Classifications: 35KO5, 32W30.
Key Words: Entire solution; heat equation; entire caloric functions; order;
           dynamics of the zeros.