On a Class of Linear Cooperative Systems with Spatio-temporal Degenerate Potentials

Abstract

This paper analyses a class of parabolic linear cooperative systems in a cylindrical domain with degenerate spatio-temporal potentials. In other words, potentials vanish in some non-empty connected subdomains which are disjoint and increase in size temporally. Then, the vanishing subdomains for the potentials are not cylindrical. Following a similar idea to the semiclassical analysis behaviour, but done here for parabolic problems, under these geometrical assumptions, the asymptotic behaviour of the system is ascertained when a parameter, in front of these potentials, goes to infinity. In particular, the strong convergence of the solutions of the system is obtained using energy methods and the theory associated with the Γ-convergence. Also, the exponential decay of the solutions to zero in the exterior of the subdomains where the potentials vanish is achieved.