On space-time periodic solutions of the one-dimensional heat equation

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June 2020, 40(6): 3997-4017. doi: 10.3934/dcds.2020037

On space-time periodic solutions of the one-dimensional heat equation

Dong-Ho Tsai ^1, and Chia-Hsing Nien ^2,

1.	Department of Mathematics, National Tsing Hua University, Hsinchu 30013, Taiwan
2.	Department of Financial Engineering, Providence University, Taichung 43301, Taiwan

Received February 2019 Published October 2019

Abstract
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We look for solutions

$ u\left( x,t\right) $

of the one-dimensional heat equation

$ u_{t} = u_{xx} $

which are space-time periodic, i.e. they satisfy the property

$ u\left( x+a,t+b\right) = u\left( x,t\right) $

for all

$ \left( x,t\right) \in\left( -\infty,\infty\right) \times\left( -\infty,\infty\right), $

and derive their Fourier series expansions. Here

$ a\geq0,\ b\geq 0 $

are two constants with

$ a^{2}+b^{2}>0. $

For general equation of the form

$ u_{t} = u_{xx}+Au_{x}+Bu, $

where

$ A,\ B $

are two constants, we also have similar results. Moreover, we show that non-constant bounded periodic solution can occur only when

$ B>0 $

and is given by a linear combination of

$ \cos\left( \sqrt{B}\left( x+At\right) \right) $

and

$ \sin\left( \sqrt{B}\left( x+At\right) \right). $

Keywords: Heat equation, time-periodic solution, space-periodic solution, space-time periodic solution, travelling wave solution.

Mathematics Subject Classification: Primary: 35K05; Secondary: 35B10.

Citation: Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037

References:

[1]	J. R. Cannon, The One-Dimensional Heat Equation, Encyclopedia of Mathematics and its Applications, 23. Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984. doi: 10.1017/CBO9781139086967.
[2]	D. V. Widder, The Heat Equation, Pure and Applied Mathematics, Vol. 67. Academic Press, New York-London, 1975.

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References:

[1]	J. R. Cannon, The One-Dimensional Heat Equation, Encyclopedia of Mathematics and its Applications, 23. Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984. doi: 10.1017/CBO9781139086967.
[2]	D. V. Widder, The Heat Equation, Pure and Applied Mathematics, Vol. 67. Academic Press, New York-London, 1975.

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