Oscillatory properties of the solutions of nonlinear delay hyperbolic differential equations of neutral type.
The authors study oscillatory properties of the solutions of nonlinear delay hyperbolic differential equations of neutral type with multi-delays under two type boundary conditions, \[ {\partial^2 \over \partial t^2}[u(t,x)+p(t)u(t-\mu (t),x )]=a(t)h(u)\triangle u+\sum_{i=1}^{m}a_i(t)h_i(u(t-\tau_i (t),x ))\triangle u(t-\tau_i(t),x)- \] \[ -\sum_{j=1}^{n}b_j(t,x)f_j(u(t-\sigma_j (t),x )), \;\;(t,x)\in \mathbb R_+\times \Omega \equiv G. \] Two different cases of boundary problems are considered. The first case is the problem under nonlinear boundary condition, \(\partial u/\partial n =g (t, x, u)\), and the second is \(\partial u/\partial n + cu=0\), on \( \mathbb R_+\times \partial\Omega \), where \(\Omega \) is a bounded domain in \(\mathbb R^N\) with a smooth boundary \(\partial \Omega \), \(\mathbb R_+=[0,\infty )\), \(\triangle \) the Laplacian operator in \(\mathbb R^N\) and \(n\) is the unit outward normal vector on \(\partial\Omega \). It is assumed that \(a,a_i,b_j,g,h^{\prime }(u),h_i^{\prime }(u),f_j(u),\tau_i,\sigma_j,\mu \) are continuous functions in some domains, \(0<\tau_i(t)<\tau \), \(0<\sigma_j(t)<\sigma \), \(0\leq \mu (t)\leq \mu \) (\(c>0,\tau,\sigma,\mu \) are constants, \(i=1,\dots ,m\), \(j=1,\dots ,n\)), \(-1\leq p(t)<0\) is two times differentiable with respect to \(t\) in \(\mathbb R_+\), \(uh(u)g(t,x,u)<0\), \(uh_i(u)g(t,x,u)<0\), \(uh_i^{\prime }(u)\geq 0\), \(uh^{\prime }(u)\geq 0\), \(f_j(u)/u\geq C_j= \text{const} >0\) (\(u\neq 0\)). The main result is that all solutions of the problems under consideration are oscillating in \(G\). Several modified statements for existence of oscillations follow from the main result. The results fully indicate that the oscillations are caused by delay and hence reveal the difference between these equations and those equations without delay. Two interesting examples illustrate the theory.
scientific article; zbMATH DE number 1880780