Oscillation for systems of nonlinear neutral type parabolic partial functional differential equations (Q1364890)

The system considered is the following nonlinear system of neutral type parabolic functional differential equation \[ \begin{multlined} {\partial\over\partial t}\left[u_i(x,t)- \sum_{k=1}^n p_k(t)u_i[x,t-\sigma_k(t)]\right]+ A_i(x,t)u_i(x,t)+ \sum_{j=1}^m B_{ij}(x,t)f_{ij}[u_j(x,t-\tau)]\\ =C_i(t)\Delta u_i(x,t)+ \sum_{j=1}^m D_{ij}(t)\Delta u_j(x,t-r_j), \end{multlined} \] where \(i=1,\ldots, m\); \((x,t)\in\Omega\times(0,\infty)\), \(\Omega\) is a bounded domain in \({\mathbb{R}}^n\) with piecewise smooth boundary, \(\Delta\) is the Laplace operator, with boundary conditions \[ {\partial u_i\over\partial n}+g_i(x,t)=0,\quad (x,t)\in\partial\Omega\times[0,\infty). \] A solution \(u\) of this problem is said to be oscillatory if at least one component \(u_i\) of \(u\) is such that for any \(\beta>0\) there exist a zero of \(u_i\) in \(\Omega\times(\beta,\infty)\). The authors give sufficient conditions for all solutions to be oscillatory.