Asymptotic behavior of positive solutions for heat equation with nonlinear term

We study the nonlinear parabolic equation u − uϕ(., u) − ∂u ∂t = 0, in Rn × (0,∞) with boundary condition u(x, 0) = u0(x), not necessarily bounded function. The nonlinearity ϕ((x, t ), u) is required to satisfy some conditions related to the parabolic Kato class P∞(Rn) while allowing existence of positive solutions of the equation and continuity of such solutions. Our approach is based on potential theory tools. © 2006 Elsevier Inc. All rights reserved.