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

We analyse the dynamics of the non-autonomous nonlinear reaction-diffusion equationu_t - Δ 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) | u |² + D (t, x) | u |, where the linear evolution operator associated with Δ + 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 - Δ u = C (t, x) u between different L^p spaces.