On a parabolic problem with nonlinear term in a half space and global behavior of solutions

We take Lip the existence and global behavior of positive continuous solutions of the following nonlinear parabolic equation Delta u - u phi(.,u) - partial derivative u/partial derivative t = 0 in R-+(n) x (0, infinity) (n >= 2) with boundary conditions u = 0 on partial derivative R-+(n) x (0, infinity) and u(x, 0) = u(0)(x). The nonlinear term is required to satisfy some conditions related to a functional class P infinity(R-+(n)), which we introduce in this paper and will be called parabolic Kato class in the half space. Our approach is based on potential theory. (C) 2007 Elsevier Inc. All rights reserved.