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Uniform attractors for a strongly damped wave equation with linear memory. (English) Zbl 0936.35037

The authors consider a strongly damped wave equation with linear memory term:

u tt (t)-αΔu t (t)-βΔu(t)- 0 μ(ζ)Δη t (ζ)dζ+gu ( t )=f(t)(1)

on Ω×(τ,) with u(x,t)=0 on Ω×, where u(0,t)=u 0 (x,t) for tτ and η t (x,s)=u(x,t)-u(x,t-s). A variety of conditions are imposed on the data μ(ζ), g(u), f(t). After some preparations, (1) is transformed into a system:

z t =Lz+N(z),z=(u,v,η),v=u t ,z(x,t)=0onΩ×,z(x,τ)=z 0 (x)·(2)

System (2) is then cast into a functional frame. With A=-Δ, the Dirichlet Laplacian, fractional powerspaces V 2σ =dom(A σ ), , 2σ are introduced for σ in the usual way. A weighted Hilbert space L μ 2 (R + ,V σ ) is then defined via φ,ψ σ,μ = 0 μ(ζ)φ(ζ),ψ(ζ) σ dζ. The phase space for (2) then is 𝒱 σ =V 1+σ ×V σ ×L μ 2 ( + ,V σ ).

A function hL loc p (,V 0 ) (V 0 =L 2 (Ω)) is called translation invariant if the closure H(h) of all translates h(·+τ)=h τ , τ is compact in L loc p (,V 0 ). Finally, a function space T p is defined, consisting of the fL loc p (,V 0 ) with norm sup( τ τ+1 f(y) p dy) 1/p <.

MSC:
35B41Attractors (PDE)
35L70Nonlinear second-order hyperbolic equations
35L20Second order hyperbolic equations, boundary value problems