Global Attractors For Fractional Damped Wave Equations In Locally Uniform Spaces

In this paper,we focus on the dynamical behavior of the solutions of nonlinear fractal strongly damped wave equations with critical growth rate in locally uniform space:utt+?ut+?(-?)?ut-?u+?(u)=f,x?RN,t>0.Where N ? 3;a and ? are positive constants;?(-?)?utis the strongly damped term,??(0,1];the external force f ? Llu2(RN);u(x,t):RN × R+?R is an unknown function;the nonlinear term ??C1(R,R)with the critical growth condition 1 + 4?/(N-2?).Well-posedness and long-time behavior for analogous equations on bounded domain have been investigated by many authors in recent years.The existence of global attractor and exponential attractor as well as the fractal dimension of attractor have been proved on bounded domain in many papers.In the case where the domain is unbounded,some compactness results are not available,so it seems hard to show the existence of global attractor directly through the existence of compact attracting set.Besides that,the general Sobolev spaces exclude traveling waves and constant functions.In order to make these special solutions included in the attractor,a number of authors think about bounded and uniformly continuous function spaces and weighted spaces,but the weighted spaces neglect some features of the solutions away from the origin of the coordinate and for which the general Sobolev embeddings are unavailable.Later,some scholars solve this problem by using locally uniform spaces which possess appropriate nesting properties and compact embeddings formula.Moreover,the locally uniform spaces contain constant functions.However,due to some differences between the embedding formula,the approach which applied to the bounded domain cannot be used in locally uniform spaces directly.It requires a completely different approach.Yang M.H.and Sun C.Y.have studied the global well-posedness and asymptotic regularity of the solutions to the strongly damped wave equations on unbounded domain.Also,they have proved the existence of global attractors of the equations in local uniform spaces.The aim of this paper is to extend this result to the case of semilinear fractionally damped wave equations with super-cubic growth rate on unbounded domain.Recently,the dynamic properties of fractional differential equations are becoming a hot topic of mathematicians and engineers.The theory of fractional calculus provides an excellent tool for the description of memory and hereditary properties,which are widely used in the fields of physics and engineering,such as fluid mechanics,biology,chemistry,material science and so on.Wave equations with fractional damping arise when waves propagate through a lossy medium,for example fractal rock layers,human tissues,different biomedical materials.As far as I know,well-posedness and attractor theory in the case of cubic non-linearity can be done similarly to the case of ? B 0,but there are not such results in the case of super-cubic non-linearity.In the process of proving the global well-posedness,we prove the asymptotic regularity of the solutions firstly,then we prove a slightly stronger attractability,the existence of(Hlu1(RN)×Llu2(RN),Hlu3(RN)× Hlu1(RN)-attractor and(Hlu1(RN)× Llu2(RN),H?1(RN)×H??(RN))-attractor.