Existence of global solutions and attractors for the parabolic equation with critical Sobolev and Hardy exponent in R^sup N

We consider asymptotic behavior for a parabolic equation (p-heat equation) with (sub-)critical Sobolev and Hardy exponent potential ... where 1 < p < N, (N ≥ 2). If V(x) ≡ λ and s = q = p, it is shown by Azorero and Alonso (1998) that the solution for such Cauchy problem depends strongly on and the best constant λN,p in Hardy’s inequality, specifically, the local or global solutions for this problem depend on the different cases of 1 < p < 2, p ≥ 2 and λ > λN,p, λ < λN,p. By using the standard domain expansion technique and some convenient integrability conditions on the weighted function V(x), we first establish that the Cauchy problem has at least one global solution for every 1 < p < N. We then apply the theory of multivalued semigroups or multivalued semiflows to get L2 (ℝN) global attractor for the p-heat equation with 2 ≤ p < N. Furthermore, the global attractor also belongs to L2 (ℝN) ∩ Lα (ℝN) when is suitably large. ProQuest: ... denotes formulae omitted.