In this paper we study the partial differential equation with piecewise constant argument of the form : \[ \begin{array}{lll} x_t(t,s)=&A(t)x(t,s)+B(t,s)x([t],s)+C(t,s)x(t,[s])+\\[0.5cm] &D(t,s)x([t],[s])+f(x(t,[s])),\ \ t,s\in \R^{+}=(0,\infty) \end{array} \] where $A(t)$ is a $k\times k$ invertible and continuous matrix function on $\R^{+}$, $B(t,s)$, $C(t,s)$, $D(t,s)$ are $k \times k$ continuous and bounded matrix functions on $\R^{+}\times \R^{+}$, $[t]$, $[s]$ are the integral parts of $t,s$ respectively and $f:\R^k\rightarrow \R^k$ is a continuous function. More precisely under some conditions on the matrices $A(t)$, $B(t,s)$, $C(t,s)$, $D(t,s)$ and the function $f$ we investigate the asymptotic behaviour of the solutions of the above equation. \end{abstract}