Longtime Dynamics Of Three Classes Of Nonlinear Wave Equations With Damping

In this paper,we investigate the well-posedness,existence and stability of attractors for three non-autonomous nonlinear damping equations as follows.（ⅰ）Firstly,we investigate the well-posedness of global solutions,the existence of pullback attractors and pullback exponential attractors and their continuity for a non-autonomous extensible beam equation with rotational inertia and nonlocal energy damping:（1-αΔ）u_tt+Δ²u-φ（‖▽u‖²）Δu-M（‖ξu‖_H²）Δu_t+f（u）=f（x,t）,where ‖ξ_u‖_H²=‖Δu‖²+‖u_t‖²,α∈[0,1]is a rotational coefficient.We show that when the growth exponent p of the nonlinearity f（u）is up to the range:1≤p≤p’=（N+2）/（N-4）⁺,the problem is well-posed.In particular,when 1≤p<p’,we prove the existence of pullback attractor and pullback exponential attractor for the related process,upper and lower semi-continuity of the family of pullback attractors on α,and Holder continuity of the family of pullback exponential attractors on α;（ⅱ）We give the global well-posedness and regularity result of the following non-autonomous extensible beam equation with nonlocal energy damping:u_tt+Δ²u-κφ（‖▽u‖²）Δu+M（‖ξ_u‖_H²）（-Δ）^αut+f（u）=h（x,t）,where κ∈[0,1],α∈[1,2),the nonlinearity f（u）is of optimal subcritical growth 1≤p<p^*=（N+4）（N-4）⁺.And we give the definitions of strong pullback attractor and strong pullback exponential attractor,that is,the compactness,the attractiveness and the finiteness of the fractal dimension of the attractors are all in the topology of the stronger space,prove the existence of strong pullback attractor and pullback exponential attractor.Moreover,we investigate the stability of pullback attractors and pullback exponential attractors on the perturbed parameterκ and dissipation index α in the topology of stronger space;（ⅲ）We investigate the well-posedness,the existence and the regularity of the time-dependent global attractor for a viscoelastic wave equation in Ω （?）R³:|（?）_tu|ρ（?）_ttu-（?）_ttΔu-h_t（0）Δu-∫₀^∞（?）_sh_t（s）Δu（t-s）ds+f（u）=g with time-dependent memory kernel which is used to model aging phenomena of the material.By establishing some delicate integration estimates along the trajectory of the solutions in the time-dependent phase space,we show that whenρ∈(1,4]and the growth exponent p of f（u）is up to the critical range 1≤p≤5,the model is well-posed.Especially,when ρ ∈（1,4）and 1≤p<5,the related process has an invariant time-dependent global attractor which has optimal regularity.