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

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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). $

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


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  • [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. WidderThe Heat Equation, Pure and Applied Mathematics, Vol. 67. Academic Press, New York-London, 1975. 
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