Non-dissipative system as limit of a dissipative one : comparison of the asymptotic regimes

Let ⊂ Rn be a bounded smooth domain in Rn. Given u0 ∈ L2(), g ∈ L∞() and λ ∈ R, consider the family of problems parametrised by p 2, ⎧ ⎪⎨ ⎪⎩ ∂u ∂t − pu = λu + g, on (0,∞) × , u = 0, in (0,∞) × ∂, u(0, ·) = u0, on , where pu := div |∇u| p−2∇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