Publication: Pullback attractors and extremal complete trajectories for non-autonomous reaction-diffusion problems
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Publication Date
2007
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Elsevier
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Abstract
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)u
between different Lp spaces.