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

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

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

 Citation:

•  [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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