Oscillation criteria for nonlinear inhomogeneous hyperbolic equations with distributed deviating arguments (Q1917538)

We consider the following nonlinear inhomogeneous hyperbolic equation with distributed deviating arguments: \[ \begin{aligned} {\partial^2\over \partial t^2} [u+ \lambda(t) u(x, t- \tau)] & + \int^b_a p(x, t, \xi) F(u(x, g[t, \xi])) d\sigma(\xi)\tag{E}\\ & = a(t) \Delta u+ f(x, t),\quad (x, t)\in G,\end{aligned} \] where \(G= \Omega\times \mathbb{R}_+\), \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with piecewise smooth boundary \(\partial \Omega\), \(\mathbb{R}_+= [0, + \infty)\), \(p\in C[\overline\Omega\times \mathbb{R}_+\times J, \mathbb{R}_+]\), \(J= [a, b]\), \(F\in C[\mathbb{R}, \mathbb{R}]\), \(a\in C[\mathbb{R}_+, \mathbb{R}_+]\), \(\lambda\in C^2[\mathbb{R}_+, \mathbb{R}]\), \(\tau\) is a constant, \(f\in C[\Omega\times \mathbb{R}_+, \mathbb{R}]\), \(g\in C[\mathbb{R}_+\times J, \mathbb{R}]\), \(\sigma\in [J, \mathbb{R}]\), and the integral in (E) is a Stieltjes integral. Throughout this paper we assume that \(g(t, \xi)\) is nondecreasing in \(t\) and \(\xi\) respectively, with \(g(t, \xi)< t\) for any \(\xi\) and \(\lim_{t\to +\infty} \inf_{\xi\in J} g(t, \xi)= + \infty\), and that \(\sigma(\xi)\) is nondecreasing in \(\xi\). We consider two kinds of boundary conditions: \[ {\partial u\over \partial N}+ \gamma(x, t) u= \mu(x, t),\quad (x, t)\in \partial \Omega\times \mathbb{R}_+,\tag{B1} \] and \[ u= \phi(x, t),\quad (x, t)\in \partial \Omega\times \mathbb{R}_+,\tag{B2} \] where \(N\) is the unit outnormal vector to \(\partial \Omega\), \(\gamma\in C[\partial \Omega\times \mathbb{R}_+, \mathbb{R}_+]\), \(\mu, \phi\in C[\partial \Omega\times \mathbb{R}_+, \mathbb{R}]\). The objective of this paper is to study the oscillatory properties of solutions of equation (E) subject to boundary conditions (B1) and (B2).