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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

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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[2] D. V. Widder, The Heat Equation, Pure and Applied Mathematics, Vol. 67. Academic Press, New York-London, 1975.   Google Scholar

show all references

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.  Google Scholar

[2] D. V. Widder, The Heat Equation, Pure and Applied Mathematics, Vol. 67. Academic Press, New York-London, 1975.   Google Scholar
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