On the Galerkin Method for Semilinear Parabolic-Ordinary Systems

We consider a general system of n_1 semilinear parabolic partial differential equations and n_2 ordinary differential equations, with locally Lipschitz continuous nonlinearities. We analyse the well-posedness of this problem, exploiting the tools of the semigroups theory, and derive other further regularity results and conditions for the boundedness of the solution. We define the Galerkin semidiscrete approximation to the system and derive optimal order error estimates in L^2 norm, under various assumptions on the nonlinear terms, on the finite dimensional subspaces in which the approximation is sought and on the regularity of the exact solution. As a byproduct, we can also show that the approximate solution is globally defined and bounded.