%0 Journal Article
%A Rodríguez Bernal, Aníbal
%A Vidal López, Alejandro
%A Robinson, James C.
%T Pullback attractors and extremal complete trajectories for non-autonomous reaction-diffusion problems
%D 2007
%@ 0022-0396
%U https://hdl.handle.net/20.500.14352/49691
%X We analyse the dynamics of the non-autonomous nonlinear reaction–diffusion equation ut −_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 _ + 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 − _u = C(t, x)ubetween different Lp spaces.
%~