Oscillation for systems of nonlinear neutral type parabolic partial functional differential equations

Abstract

This paper discusses the oscillation of solutions for systems of nonlinear neutral type parabolic partial functional differential equations of the form $$\frac{\partial }{{\partial t}}[u_i (x,t) - \sum\limits_{k = 1}^n {p_k (t)u_i (x,t - \sigma _k (t))] + A_i (x,t)u_i (x,t) + \sum\limits_{j = 1}^m {B_{ij} (x,t)f_{ij} (u_j (x,t - \tau )) = C_i (t)\Delta u_i (x,t) + \sum\limits_{j = 1}^m {D_{ij} (t)\Delta u_i (x,t - r_j ),i = 1,2,...,m,(x,t) \in \Omega {\text{ x }}(0,\infty ) \equiv G} } } $$ (1) where Ω is a bounded domain in Rn with piecewise smooth boundary. $$\Delta u_1 (x,t) = \sum\limits_{j = 1}^n {\left( {\frac{\partial }{{\partial x_j^2 }}^2 } \right)u_1 (x,t),\tau ,r_j = {\text{ const}}{\text{.}} > 0} $$ (1) Sufficient conditions are obtained for oscillation of solutions of the systems, these results are illustrated by some examples.