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 by-product, we can also show that the approximate solution is globally defined and bounded.