Oscillation for Certain Nonlinear Neutral Partial Differential Equations

Abstract

We present some new oscillation criteria for second-order neutral partial functional differential equations of the form $(\partial /\partial t)\left\{p\right(t\left)\right(\partial /\partial t\left)\right[u(x,t)+{\sum }_{i=1}^l{\lambda }_i\left(t\right)u(x,t-{\tau }_i)\left]\right\}=a\left(t\right)\Delta u(x,t)+{\sum }_{k=1}^s{a}_k\left(t\right)\Delta u(x,t-{\rho }_k(t\left)\right)-q(x,t)f\left(u\right(x,t\left)\right)-{\sum }_{j=1}^m{q}_j(x,t)f_j\left(u\right(x,t-{\sigma }_j\left)\right)$, $(x,t)\in \Omega \times R^{+}\equiv G$, where $\Omega$ is a bounded domain in the Euclidean $N$-space $R^N$ with a piecewise smooth boundary $\partial \Omega$ and $\Delta$ is the Laplacian in $R^N$. 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.