On the Dynamics of Abstract Retarded Evolution Equations

Desheng Li,Jinying Wei,Jintao Wang

doi:10.1155/2013/359310

Desheng Li, Jinying Wei + Show 1 more

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https://doi.org/10.1155/2013/359310

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Abstract

This paper is concerned with the dynamics of the following abstract retarded evolution equation: in a Hilbert space , where is a self-adjoint positive-definite operator with compact resolvent and is a locally Lipschitz continuous mapping. The dissipativity and pullback attractors are investigated, and the existence of locally almost periodic solutions is established.

Highlights

This paper is concerned with the abstract retarded evolution equation d dt u (t) + Au = F (u (t − r1), u rn)) g (1)in a Hilbert space H, where A : D(A) ⊂ H → H is a selfadjoint positive-definite operator with compact resolvent, F : D(Aα)n → H (α ∈ [0, 1/2]) is a locally Lipschitz continuous mapping with at most a linear growth rate, g ∈ C(R; H) is a bounded function, and r1, . . . , rn ≥ 0 are constant time lags
By the assumptions on F we know that G0 : [0, σ] × Hα → H is a continuous mapping which is locally Lipschitz in V
By repeating some argument as above, one can obtain an extension of u on some larger interval [−r, T + δ)

Summary

Introduction

This paper is concerned with the abstract retarded evolution equation d dt u (t). in a Hilbert space H, where A : D(A) ⊂ H → H is a selfadjoint positive-definite operator with compact resolvent, F : D(Aα)n → H (α ∈ [0, 1/2]) is a locally Lipschitz continuous mapping with at most a linear growth rate, g ∈ C(R; H) is a bounded function, and r1, . . . , rn ≥ 0 are constant time lags. There have been a lot of works in this line for some types of retarded partial differential equations and the abstract equations as in (1) (see, e.g., [3,4,5,6,7,8]), due to some strict restrictions on the nonlinearities, we find that the known results in most of the existing works do not apply to many important PDE examples as the parabolic one given in Section 5 whose nonlinearity involves the gradient ∇u of the unknown function In this present work, we will try to establish some new results in more regular spaces under weaker assumptions.

Preliminaries

Pullback Attractors of Cocycles

Locally Almost Periodic Solutions

An Example