Existence of global solutions and attractors for the parabolic equation with critical Sobolev and Hardy exponent in RN. (August 2018) - Qian, Chenyin; Shen, Zifei
Abstract: We consider asymptotic behavior for a parabolic equation ( p-heat equation ) with (sub-)critical Sobolev and Hardy exponent potential u t − div ( | ∇ u | p − 2 ∇ u ) + f ( u ) = V ( x ) | u | q − 2 | x | s u, 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 p and the best constant λ N, 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 > λ N, p, λ < λ N, 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 L 2 ( R N ) global attractor for the p-heat equation with 2 ≤ p < N . Furthermore, the global attractor also belongs to L 2 ( R N ) ∩ L α ( R N ) when N is suitably large.