Global existence and asymptotic behavior of solutions for nonlinear parabolic equations on unbounded domains

We study the existence and the asymptotic behavior of positive solutions for the parabolic equation a u − t u + Vup =0 on D × (0,∞), where a>0, D is a some unbounded domain in Rn, n 3 and V belongs to a new parabolic class J∞ of singular potentials generalizing the well-known parabolic Kato class at infinity P∞ introduced recently by Zhang. We also show that the choice of this class is essentially optimal. © 2005 Published by Elsevier Inc.