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). $