The consensus problem for a multi-agent system composed of agents described by semilinear or quasilinear parabolic partial differential equations is solved. For simplicity, only the Dirichlet boundary conditions are considered, however, the proposed method is applicable to systems with Neumann boundary conditions as well. The essence of the method is similar to the exact feedback linearization that has been successfully used to solve the consensus problem for nonlinear multi-agent systems. An auxiliary linear system is defined by subtracting the nonlinear terms from the original control signal; this system is thus composed of agents described by linear parabolic partial differential equations. Then, the consensus problem for the linear multi-agent system is solved. Finally, the original control is recovered; it is shown that this transformation of the input does not cause a loss of the synchronization, hence, the synchronization of the multi-agent system is rigorously proved. Robustness issues originating from imprecise description of the system are also thoroughly studied. Numerical aspects of the proposed algorithm are also discussed. The method to achieve the theoretical results is the Galerkin approximation of the original partial differential equation which allows to convert the original partial differential equation into a set of ordinary differential equations. The estimates of the numerical error caused by this approximation are derived. The results are illustrated by an example.