Electron. J. Differential Equations, Vol. 2021 (2021), No. 44, pp. 1-25.

Entire solutions for the heat equation

Vassilis G. Papanicolaou, Eva Kallitsi, George Smyrlis

Abstract:
We consider the solutions of the heat equation

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

form a discrete set in $\mathbb{C}^2$ and, then, we derive the t-evolution equations of zk(t). These are differential equations that hold for all but countably many ts in $\mathbb{C}^2$.

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.
DOI: https://doi.org/10.58997/ejde.2021.44

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Vassilis G. Papanicolaou
National Technical University of Athens
Zografou Campus, 157 80, Athens, Greece
email: papanico@math.ntua.gr
Eva Kallitsi
National Technical University of Athens
Zografou Campus, 157 80, Athens, Greece
email: evpapa@hotmail.com
George Smyrlis
National Technical University of Athens
Zografou Campus, 157 80, Athens, Greece
email: gsmyrlis@math.ntua.gr

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