We design a network-based H∞ filter for semilinear diffusion partial differential equations over a rectangular domain under distributed in space measurements. The sampled in time measurements are sent to the observer over a communication network. The objective is to enlarge the sampling time intervals, while preserving a satisfactory error system performance in the presence of variable network-induced delays. We suggest to divide the domain into a finite number of rectangular sub-domains, where sensing devices provide spatially averaged state measurements to be transmitted through communication network. Sufficient conditions in terms of Linear Matrix Inequalities (LMIs) for the internal exponential stability and L2-gain analysis of the estimation error are derived via the time-delay approach to networked control systems and the direct Lyapunov-Krasovskii method. Numerical example illustrates the efficiency of the method.