Oscillation of hyperbolic equations with functional arguments (Q1208324)

The oscillation of hyperbolic pde's with functional arguments of the form \[ {\partial^ 2 u\over \partial t^ 2}+ p(x,t)u(x,t)+\sum^ k_{i=1} p_ i(x,t)u(x,\tau_ i(t))= a(t)\Delta u+ \sum^ m_{j=1} a_ j(t)\Delta u(x,\sigma_ j(t))\tag{1} \] is considered. First, the problem is reduced, by avaraging, to the oscillation of ordinary delay inequalities of the form \[ y''(t)+ q(t)y(t)+ \sum^ n_{i=1} q_ i(t)y(g_ i(t))\leq 0. \] It turns out that all solutions of (1) oscillate if, in addition to some continuity conditions, we have \[ \liminf_{t\to\infty} \int^ t_{\tau_ i(t)} p_ i(s)\tau_ i(s)\exp\left(\int^ s_{\tau_ i(s)} rp(r)dr\right) ds>{1\over e} \] for some \(i\in \{1,\dots,k\}\), where \(p(t)= \min_ x p(x,t)\), \(p_ i(t)= \min_ x p_ i(x,t)\).