Oscillation for certain nonlinear neutral partial differential equations
Summary: We present some new oscillation criteria for second-order neutral partial functional differential equations of the form \((\partial/\partial t)\{p(t)(\partial/\partial t)[u(x,t)+ \sum_{i=1}^l \lambda_i(t)u(x,t-\tau_i)]\}= a(t)\Delta u(x,t)+ \sum_{k=1}^s a_k(t) \Delta u(x,t-\rho_k(t))- q(x,t)f(u(x,t))- \sum_{j=1}^m q_j(x,t)f_j(u(x,t-\sigma_j))\), \((x,t)\in\Omega\times\mathbb R^+\), where \(\Omega\) is a bounded domain in the Euclidean \(N\)-space \(\mathbb R^N\) with a piecewise smooth boundary \(\partial\Omega\) and \(\Delta\) is the Laplacian in \(\mathbb R^N\). Our results improve some known results and show that the oscillation of some second-order linear ordinary differential equations implies the oscillation of relevant nonlinear neutral partial functional differential equations.
scientific article; zbMATH DE number 5816465