Universal Attractor For Semilinear Wave Equations With Fractional Damping On Unbounded Domain

In this paper,we investigate the asymptotic behavior of solutions for the following semilinear wave equations with fractional damping:(?)When the nonlinearity is subcritical case,we prove the existence of an universal attractor for the above equation.The longterm behavior of the weakly(or strongly)damped wave equations on bounded domains has been studied by many authors,where the existence of the universal attractor of finite Hausdorff and fractal dimensions has been proved.Concerning the study of the asymptotic behavior of wave equations on unbounded domains,the existing literature is so vast.The main difficulty here is represented by the fact that,for an unbounded domain,the embedding H~1(R~3)?L~6(R~3)is continuous but no longer compact,therefore one cannot exploit directly compactness results to show,for instance,the existence of compact attracting sets.In recent years,wave equations with fractional damping term have attracted more and more attention.From an applied point of view this is related to the fact that such equa-tions model various processes with frequency depending attenuation.Frequency-dependent attenuation has been observed in a wide range of important engineering areas such as a-coustics,viscous dampers in seismic isolation of buildings,structural vibration,seismic wave propagation,anomalous diffusions occurring in porous media,just to mention a few.From a mathematical point of view,even in linear case(g?0)properties of such equations demonstrate non-trivial dependence on ?.In the presence of the non-linear term of type g(u)?u|u|?,even well-posedness of the above problem becomes questionable.In the weakly damped case Feireisl has studied the existence of global attractors of wave equations with the decomposition method of solutions.Hi is based on the decomposition of the solution u = v +w,where v is asymptotically small,and w belongs to a compact set for all times.Feireisl's technique relies on the fact that,due to the hyperbolicity,the propagation speed of initial disturbances is finite.In our situation,Feireisl's approach does not work.Indeed,if from one side the damping term(-?)~?u_t increases the dissipation,from the other gives the equation a partially parabolic character.Hence the equations have more regularization,but also an infinite propagation speed of initial disturbances.We are therefore compelled to introduce a new argument,namely,besides decomposing the solution as u = v + w,we need a further decomposition of the term w to get the desired compactness.Hence,we will proceed with a decomposition of the solution,using a suitable cutoff,in order to obtain a family of nested sets,which turn out to be compact and almost attracting.