Poincare's inequality and global solutions of a nonlinear parabolic equation

We study the equation u(t) - Delta u = u(p) - mu\del u\(q), t greater than or equal to 0 in a general (possibly unbounded) domain Omega subset of IRN. When q greater than or equal to p, we show a close connection between the Poincare inequality and the boundedness of the solutions, To be more precis e, if q > p (or q = p and mu large enough), we prove global existence of al l solutions for any domain Omega where the Poincare inequality is valid, Wh en mu is large enough, all solutions are bounded and decay exponentially to zero. Conversely, if Omega contains arbitrarily large balls (if N less tha n or equal to 2 and Omega is finitely connected, this means precisely that the Poincare inequality does not hold), then there always exist unbounded s olutions. Moreover, if Omega = IRN, there exist global solutions which blow -up at every point in infinite time. Various qualitative properties of the solutions are also obtained. (C) Elsevier, Paris.