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Non-autonomous nonlocal partial differential equations with delay and memory. (English) Zbl 1451.35072

Summary: The paper addresses a kind of non-autonomous nonlocal parabolic equations when the external force contains hereditary characteristics involving bounded and unbounded delays. First, well-posedness of the problem is analyzed by the Galerkin method and energy estimations in the phase space \(\mathbf{C}_\rho(X)\). Moreover, some results related to strong solutions are proved under suitable assumptions. The existence of stationary solutions is then established by a corollary of the Brower fixed point theorem. By constructing appropriate Lyapunov functionals in terms of the characteristic delay terms, a deep analysis on stability and attractiveness of the stationary solutions is established. Furthermore, the existence of pullback attractors in \(L^2(\Omega)\), with bounded and unbounded delays, is shown. We emphasize that, to prove the existence of pullback attractors in the unbounded delay case, a new phase space, \(E_\gamma\), has to be constructed.

MSC:

35K58 Semilinear parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35R10 Partial functional-differential equations
35B41 Attractors
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References:

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