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On a system of Klein-Gordon type equations with acoustic boundary conditions. (English) Zbl 1060.35118
This paper deals with the existence, uniqueness and asymptotic behaviour of solutions to the \((k\times k)\) system of Klein-Gordon type equations \[ \begin{aligned} u_1^{\prime\prime} &- \Delta_x u_1+ \alpha_1 u_1+ a_{12} u_1 u^2_2+ a_{13} u_1u^2_3+\cdots+ a_{1k} u_1 u^2_k= f_1,\\ u^{\prime\prime}_k&- \Delta_x u_k+ \alpha_k u_k+ a_{k1} u_k u^2_1+ a_{k2} u_k u^2_2+\cdots+ a_{k(k-1)} u_k u^2_{k-1}= f_k, \end{aligned}\tag{1} \] where \(\alpha_i\) \((1\leq i\leq k)\), \(a_{ij}= a_{ji}\) are nonnegative constants, \(a_{ii}= 0\). The authors prove the existence of global weak solutions to (1) and the exponential decay of the energy for \(f_i\equiv 0\) and suitable boundary and initial conditions.

MSC:
35Q40 PDEs in connection with quantum mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35B40 Asymptotic behavior of solutions to PDEs
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