Oscillatory solutions of delay hyperbolic system with distributed deviating arguments (Q1883207)

The authors study a system of partial differential equations of hyperbolic type with delay and continuous distributed deviating arguments \[ \begin{multlined} \partial_t[\partial_t(u_i(x,t)+\sum_{k=1}^{n}C_k(t)u_i(x,t-\tau_k))]=a_i(t)\Delta u_i(x,t) + \\ +\sum_{j=1}^{m}b_{ij}(t)\Delta u_j(x,t-\eta )-p_i(x,t)u_i(x,t)-\sum_{k=1}^{n}\int_{a}^{b}q_{ij}(x,t,\xi )f_{ij}[u_j(x,g(t,\xi ))]\,d\sigma (\xi ), \end{multlined} \] \((i=1,2,\dots ,m)\) under boundary conditions, \[ \partial_{\nu }u_i+\varphi _i(x,t)u_i=0, \;\;(x,t)\in \partial\Omega \times \mathbb R_+, \;\;i=1,2,\dots ,m, \] \[ u_i=0, \;\;(x,t)\in \partial\Omega \times \mathbb R_+, \;\;i=1,2,\dots ,m. \] Here \(\Omega \) is a boundary domain in \(\mathbb R^N\) with a piecewise continuous smooth boundary, \(\mathbb R_+ = [0,+\infty )\), \(\Delta \) is the Laplacian in \(\mathbb R^N\), \(\nu \) denotes the unit exterior vector normal to \(\partial \Omega \), \(\varphi_i(x,t)\in C(\partial \Omega \times \mathbb R_+,\mathbb R_+)\) \((i=1,2,\dots ,m)\). The integrals of the considered equations are assumed to be in the Stieltjes sense. Using the eigenvalue method, the authors obtain several sufficient conditions for oscillation of the solutions to the system under consideration.