I am reading *The Laplacian on A Riemannian Manifold*. In the book, author defines the heat equation on manifold. In the situation $\mathbb R$ and $\mathbb S$, people often use Fourier transformation and Fourier series to solve heat equations.


In addition, *A First Course in Harmonic Analysis* generalizes the conception of Fourier analysis in LCA group.


So my question is 



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> Is there a general method of Fourier analysis to solve the heat equation on LCA group, like $\mathbb R$ and $\mathbb S$?
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Any reference and any advice is helpful. Thank you.



Heat Equation on LCA Group
I suspect there is a reference that covers what one can say about the heat equation on a locally compact abelian group. I do not know what this would say in general - for instance what is the heat equation on $\mathbb{Z}/n\mathbb{Z}$? 


But things are known at least in the context of compact Lie groups. A reference for this is E. Stein's *Topics in Harmonic Analysis: Related to the Littlewood--Paley Theory* This is discussed briefly in the second chapter. A key tool in it is the Peter-Weyl theorem which you can think of as a non-abelian version of Fourier analysis. 


