Long Time Dynamic Behavior Of A Class Of Generalized Kirchhoff Equations

My paper is mainly discusses the long-term dynamic behavior of a class of generalized nonlinear Kirchhoff equations.Firstly,at the suitable assumptions of nonlinear source term g（u）、Kirchhoff stress term M（s）,the time-based consistent prior estimation is applied,the approximate solution of the equation is constructed by using the Galarkin finite element method,and according to the result of taking the limit of the equation,the existence and uniqueness of the overall solution is obtained.Then,a bounded absorption setB_{0 k} is obtained by prior estimation,and the Rellich-kondrachov’s compact embedding theorem is used to prove the complete continuity of the solution semigroupS（t）:E_k→E _k in the phase space E_k=H^2m+k×H^k,furthermore,we proved the equation has a family of the global attractors A_k.Then,linearize the equation as P_t+∧_εP=0,and verify that the semigroups is Frechet differentiability onE_k,and the finite-dimensional estimation of the Hausdorff dimension and Fractal dimension was obtained.Secondly,by demonstrating the properties of discrete squeezing,the existence of a family of the exponential attractor of the differential equations is obtained,according to the Hadamard’s graph transformation method,the spectral interval condition is proved to be true,therefore,we get the existence of a family of the inertial manifolds for the equation.Finally,we replace additive white noise q（x）（?）with the external force term f（x）in the initial kirchhoff equation,according to Ornstein-Uhlenbeck process,we conversion the random system,and then to estimate the solution of the system to get a bounded stochastic absorption set（?）_0k（ω）∈D（E_k）.Subsequently,according to the theorem of compact embeddingE_k（?）E₀,when t=0,P_a.e.ω∈Ω,we get that the random dynamic system{S（t,w）,t≥0}is progressively compact,and then proving a family of the random attractors that is exists.