Stability of strong attractors for the extensible beam equation with gentle dissipation

Abstract

The paper investigates the stability of the strong global and exponential attractors for the extensible beam equation with gentle dissipation: utt−βM(‖∇u‖2)Δu+Δ2u+(−Δ)2αut+f(u)=g(x), where the perturbed parameter α∈(0,1/2] is a dissipative index and β∈[0,1] is an extensibility parameter. We show that when the nonlinearity f(u) is of the optimally subcritical growth depending on α: 1≤p<pα:=N+8α(N−4)+, (i) the strong (H,H2)-global attractors obtained in [15] for this model are upper semicontinuous with respect to α in H2-topology and with respect to β in stronger V4×V2+α1-topology, with α1∈(0,α), respectively; (ii) for any fixed β∈[0,1] and α0∈(0,1/2], the related dynamical system (Sα(t),H) has a family of strong (H,H2)-exponential attractors Eexpα which are continuous with respect to α at α0 in H2-topology; (iii) for any fixed α∈(0,1/2], the related dynamical system (Sβ(t),H) has a family of strong (H,V4×V2+2α1)-exponential attractors Eexpβ which are Hölder continuous with respect to β at any point β0∈[0,1] in V4×V2+2α1-topology. All these strong global and exponential attractors are also the standard global and exponential attractors of optimal regularity. The method used here allows to obtain the stability of the strong attractors with respect to the perturbed parameters in strong topology instead of only standard energy topology.