Pullback attractors and extremal complete trajectories for non-autonomous reaction-diffusion problems

We analyse the dynamics of the non-autonomous nonlinear reaction-diffusion equation u(t) - Delta u = f(t, x, u), subject to appropriate boundary conditions, proving the existence of two bounding complete trajectories, one maximal and one minimal. Our main assumption is that the nonlinear term satisfies a bound of the form f(t, x, u)u <= C(t, x)vertical bar u vertical bar(2) + D(t, x)vertical bar u vertical bar, where the linear evolution operator associated with Delta + C(t, x) is exponentially stable. As an important step in our argument we give a detailed analysis of the exponential stability properties of the evolution operator for the non-autonomous linear problem u(t) - Delta u = C(t, x)u between different L-P spaces. (c) 2007 Elsevier Inc. All rights reserved.