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

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

 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
 $\sin\left( \sqrt{B}\left( x+At\right) \right).$
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:
  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  D. V. Widder, The Heat Equation, Pure and Applied Mathematics, Vol. 67. Academic Press, New York-London, 1975. Google Scholar

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##### References:
  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  D. V. Widder, The Heat Equation, Pure and Applied Mathematics, Vol. 67. Academic Press, New York-London, 1975. Google Scholar
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