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The attractors for weakly damped non-autonomous hyperbolic equations with a new class of external forces. (English) Zbl 1132.35018

The authors introduce for weakly dissipative problems (in particular for weakly damped non-autonomous hyperbolic equation) a new class of functions, which are more general than translation compact so far used in the study of long-time behaviour of non-autonomous equations of mathematical physics. Subsequently, they study the uniform attractors for weakly damped non-autonomous hyperbolic equations with this new class (satisfying so-called C * -condition) of time-dependent external forces g(t,x) and prove the existence of the uniform attractors for the equation

2 u t 2 +αu t-Δ x u+f(u)=g(t,x),u| Ω =0,
u(τ,x)=u τ (x), t u(τ,x)=p τ (x),α>0,

where Ω is a bounded domain in N and f, g, u τ , p τ satisfying some natural conditions.

35B41Attractors (PDE)
35B40Asymptotic behavior of solutions of PDE
58J45Hyperbolic partial differential equations on manifolds
35L70Nonlinear second-order hyperbolic equations
35L20Second order hyperbolic equations, boundary value problems