Capacity solution to a coupled system of parabolic–elliptic equations in Orlicz–Sobolev spaces

The existence of a capacity solution to a coupled nonlinear parabolic–elliptic system is analyzed, the elliptic part in the parabolic equation being of the form -\,div\, a(x,t,u,∇ u) . The growth and the coercivity conditions on the monotone vector field a are prescribed by an N-function, M, which does not have to satisfy a Δ ₂ condition. Therefore we work with Orlicz–Sobolev spaces which are not necessarily reflexive. We use Schauder’s fixed point theorem to prove the existence of a weak solution to certain approximate problems. Then we show that some subsequence of approximate solutions converges in a certain sense to a capacity solution.