zbMATH — the first resource for mathematics

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).

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
PDF BibTeX Cite
Full Text: DOI EuDML