The authors consider the following system of partial functional differential equation of neutral type \[ \begin{aligned} &\frac{\partial}{\partial t} \Biggl\{p(t)\frac{\partial}{\partial t} \Biggl[u_i(x,t) +\sum_{i=1}^d\lambda_r(t)u_i(x.t-\tau_r)\Biggr] \Biggr\}\\ &=a_i(t)\Delta u_i(x,t) +\sum_{k=1}^sa_{ik}(t)\Delta u_i(x,p_k(t))-q_i(x,t)u_i(x,t)-\\ & \sum_{h=1}^l\sum_{j=1}^mq_{ijh}(x,t)u_j(x,\sigma_h(t)),\\ &(x,t)\in \Omega \times \mathbb{R}^+=G,\;i=1,2,\cdots,m,\end{aligned} \tag{1} \] with the boundary conditions \[ \frac{\partial u_i(x,t)}{\partial n}+g_i(x,t)u_i(x,t)=0, \quad (x,t)\in \partial \Omega\times \mathbb{R}^+,\;i=1,2,\cdots,m,\tag{2} \] or \[ u_i(x,t)=0,\qquad (x,t)\in \partial \Omega\times \mathbb{R}^+,\;i=1,2,\cdots,m;\tag{3} \] where \(\Omega\subset \mathbb{R}^n\) is a bounded domain with piecewise smooth boundary \(\partial \Omega\); \(\Delta\) is the Laplacian operator; \(n\) is the unit exterior normal vector to \(\partial\Omega\); \(g_i(x,t)\) is a nonnegative continuous function on \(\partial \Omega\times \mathbb{R}^+\). If \(u=\{u_1,u_2,\cdots,u_m\}\) is a nonoscillatory solution of (1),(2) or (1),(3), then defining \(V(t)=\sum_{i=1}^mV_i(t)\) and \[ V_i(t)=\int_{\Omega}u_i(x,t)\text{ sign }u_i(x,t)dx\quad \text{or}\quad V_i(t)=\int_{\Omega}\varphi(x)u_i(x,t)\text{ sign }u_i(x,t) dx, \] where \(\varphi(x)\) is the eigenfunction of the Dirichlet problem: \[ \Delta u+\alpha u=0,\quad x\in \Omega,\quad u=0,\quad x\in \partial \Omega\tag{4} \] corresponding to the smallest eigenvalue \(\alpha_1\) of (4), it is derived that \(V(t)\) satisfies a functional differential inequality, and thus sufficient conditions for oscillation of (1),(2) or (1),(3) are obtained. One group of sufficient conditions for oscillation are \[ \lim_{t\to \infty}\int_{t_0}^t\frac{1}{p(s)}ds=+\infty,\qquad \sum_{r=1}^d\lambda_r(t)<1, \]
\[ \int^{\infty}\min_{1\leq i\leq m} \{\min_{x\in \overline{\Omega}}q_i(x,t)\} \Biggl[1-\sum_{r=1}^d\lambda_r(t) \Biggr] dt=+\infty, \] and some other assumptions on the coefficients of (1). In this article, the author says that there were only two papers on the oscillation for the system of partial functional differential equations. It needs to point out that some other papers have been published for studying this problem such as B. G. Zhang and B. S. Lalli [J. Aust. Math. Soc., Ser B 34, 245-256 (1992; Zbl 0759.39002)], P. X. Weng [Bull. Inst. Math., Acad. Sin. 24, No. 1, 33-47 (1996; Zbl 0847.35141)]. The technique and method used in this article are similar to that in the papers mentioned above.