Non-dissipative System as Limit of a Dissipative One: Comparison of the Asymptotic Regimes

Let Omega subset of R-n be a bounded smooth domain in R-n. Given u(0) is an element of L-2(Omega), g is an element of L-infinity (Omega) and lambda is an element of R, consider the family of problems parametrised by p SE arrow 2, {partial derivative u/partial derivative t - Delta(p)u = lambda u + g, on (0, infinity) x Omega, u = 0, in (0, infinity) x partial derivative Omega, u(0, center dot) = u(0), on Omega, where Delta(p)u := div(vertical bar del u vertical bar(p-2)del u) denotes the p-laplacian operator. Our aim in this paper is to describe the asymptotic behavior of this family of problems comparing compact attractors in the dissipative case p > 2, with non-compact attractors in the non-dissipative limiting case p = 2 with respect to the Hausdorff semi-distance between then.