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Finite-dimensional approximation of solutions for a nonlinear system of coupled hyperbolic equations. (Chinese. English summary) Zbl 0968.35012
The paper concerns the system of two nonlinear wave equations \[ \left.\begin{aligned} u_{tt} &-D_1\Delta u-p_1 u+p_2 uv= f(x,t)\\ v_{tt} &-D_2\Delta v- p_2 uv+ p_3v= g(x,t)\end{aligned}\right\} x\in\Omega,\;t>0 \] with homogeneous Dirichlet boundary conditions and initial conditions \(u|_{t=0}= \varphi_1\), \(v|_{t=0}= \psi_1\), \(u_t|_{t=0}= \psi_2\), \(v_t|_{t=0}= \psi_2\). The approximating solutions are constructed by means of the eigenfunctions of the eigenvalue problem \(-\Delta w= \lambda w\) in \(\Omega\), \(u= 0\) on \(\partial\Omega\).
MSC:
35A35 Theoretical approximation in context of PDEs
35L70 Second-order nonlinear hyperbolic equations
35L55 Higher-order hyperbolic systems
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