On a system of Klein-Gordon type equations with acoustic boundary conditions
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.
scientific article; zbMATH DE number 2082145