Asymptotically self-similar global solutions for Hardy-Hénon parabolic systems

Abstract

. In this paper we study the nonlinear parabolic system ∂ t u = Δ u + a | x | − γ | v | p − 1 v , ∂ t v = Δ v + b | x | − ρ | u | q − 1 u , t > 0, x ∈ R N \{ 0 } , N (cid:2) 1, a , b ∈ R , 0 (cid:3) γ < min ( N , 2 ) , 0 < ρ < min ( N , 2 ) , p , q > 1. Under conditions on the parameters p , q , γ and ρ we show the existence and uniqueness of global solutions for initial values small with respect of some norms. In partic- ular, we show the existence of self-similar solutions with initial value Φ = ( ϕ 1 , ϕ 2 ) , where ϕ 1 , ϕ 2 are homogeneous initial data. We also prove that some global solutions are asymptotic for large time to self-similar solutions.