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Oscillation criteria for nonlinear inhomogeneous hyperbolic equations with distributed deviating arguments. (English) Zbl 0852.35009
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).

##### MSC:
 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35R10 Partial functional-differential equations 35K40 Second-order parabolic systems 35L70 Second-order nonlinear hyperbolic equations
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