The multi-criteria optimization approach based on approximation and visualization of the feasible criterion set (FCS) has proven to be an effective decision support tool. In the framework of this approach, decision maker, after visually examining the Pareto frontier of the FCS, indicates the preferred feasible goal directly at the frontier. The paper is devoted to the application of the approach in the case of decision-making situations described by partial derivative equations. In such problems, the right-hand sides of the equations and the boundary conditions are the control functions that need to be selected. An example is the decision problem described by the evolutionary equation. The FCS approximation for the system under consideration is based on replacing the original system with its finite-dimensional analogue and approximating the FCS for it.