Where can I find sufficient conditions that guarantee a unique periodic solution of the linear telegraph equation?

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Nicolás Faedo

Have a look at the paper below:

https://www.sciencedirect.com/science/article/abs/pii/S0362546X97004781

The linear case is a particular case of that discussed in the paper above, by simply letting $h(x,u) = 0$ for all $x$ and $u$.

Hope this helps.

Jumaa Salman Chiad

This links have good information

https://www.hindawi.com/journals/mpe/2018/2805161

Mohammed Elamin

I think this paper is a good source:

Article Periodic solutions of parabolic and telegraph equations with...