On the oscillation of solutions of hyperbolic partial functional differential equations (Q1818180)

Solutions of the hyperbolic functional differential equation \[ \begin{multlined}{\partial^2\over\partial t^2} [u+ \lambda(t)u(x, \tau(t))]= a(t)\Delta u- c(x,t,u, u(x,\rho(t)))-\\ \int^b_a q(x,t,\xi) u[x, g(t,\xi)] d\sigma(\xi)+ f(x, t),\end{multlined} \] with Neumann boundary functions \(\varphi\), \(\psi\) are shown to oscillate under standard continuity conditions on the functions in the equation, provided the following `blow up' conditions hold: \[ \begin{aligned} & \liminf_{t\to+\infty} \int^t_{t_1}\Biggl(1-{s\over t}\Biggr)\Biggl[- a(s)\int_\Omega f(x,s) \Phi dx\Biggr] ds= -\infty,\\ & \limsup_{t\to+\infty} \int^t_{t_1} \Biggl(1-{s\over t}\Biggr)\Biggl[- a(s) \int_\Omega f(x,s) \Phi dx\Biggr] ds= +\infty\end{aligned} \] for some function \(\Phi\) where \(\Omega\) is the domain of definition of the equation.