Shadow limit for parabolic-ODE systems through a cut-off argument

We study a shadow limit (the infinite diffusion coefficientlimit) of a system of ODEs coupled with a diagonal system of semilinear heat equations in a bounded domain with homogeneous Neumann boundary conditions. The recent convergence proof by the energy approach from [19], developed for the case of a single PDE, is revisited and generalized to the case of the coupled system. Furthermore, we give a new convergence proof relying on the introduction of a well-prepared related cut-off system and on a construction of the barrier functions and comparison test functions, new in the literature. It leads to the L∞-estimates proportional to the inverse of the diffusion coefficient. This contribution is dedicated to the memory of Professor Sibe Mardešić and his professional life dedicated to the development of the mathematical sciences in Croatia.