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

We analyse the dynamics of the non-autonomous nonlinear reaction–diffusion equationut−Δu=f(t,x,u),ut−Δu=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|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)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 ut−Δu=C(t,x)uut−Δu=C(t,x)u between different LpLp spaces.