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Periodic solutions to Klein-Gordon systems with linear couplings

In this paper, we study the nonlinear Klein-Gordon systems arising from relativistic physics and quantum field theories $$\left\{\begin{array}{lll} u_{tt}- u_{xx} +bu + \varepsilon v + f(t,x,u) =0,\; v_{tt}- v_{xx} +bv + \varepsilon u + g(t,x,v) =0 \end{array}\right. $$ where $u,v$ satisfy the Dirichlet boundary conditions on spatial interval $[0, π]$, $b>0$ and $f$, $g$ are $2π$-periodic in $t$. We are concerned with the existence, regularity and asymptotic behavior of time-periodic solutions to the linearly coupled problem as $\varepsilon$ goes to 0. Firstly, under some superlinear growth and monotonicity assumptions on $f$ and $g$, we obtain the solutions $(u_\varepsilon, v_\varepsilon)$ with time-period $2π$ for the problem as the linear coupling constant $\varepsilon$ is sufficiently small, by constructing critical points of an indefinite functional via variational methods. Secondly, we give precise characterization for the asymptotic behavior of these solutions, and show that as $\varepsilon\rightarrow 0$, $(u_\varepsilon, v_\varepsilon)$ converge to the solutions of the wave equations without the coupling terms. Finally, by careful analysis which are quite different from the elliptic regularity theory, we obtain some interesting results concerning the higher regularity of the periodic solutions.