The Long Time Behavior For A Class Of High-order Kirchhoff Type Coupled Wave Equations

In this paper,the long-term behavior of the high-order Kirchhoff-coupled wave e-quations is considered.Under the appropriate assumptions,firstly applying the Galerkin method,the problem is transformed into a finite dimensional case,the approximate so?lution is constructed based on the existence theorem of the solution of the system of nonlinear ordinary differential equations,combined with a priori estimation,which proves the existence of the global solution(u(x,t),P(x,t),v(x,t),q(x,t))?L?((0,+?);E1)of this equation set,and obtains its solution is unique.At the moment,Define the solution semigroup S(t),by way of the operator semigroup,the corresponding continuous solution semigroup has a compact global attractor.Next,Adopt the thought method proposed by Eden et al.,according to the definition of projection and the construction of function,the equivalent norm is obtained,the Lipschitz continuity and the discrete squeezing property of its solution semigroup are proved successively,thus the existence of the exponential attractor for the solution semigroup is got.In addition,consider its equivalent first-order evolution equation,construct the graph model(U,V)x,and determine the eigenvalue of the matrix operator A*,further by using the Hadamard graph transformation method,the existence of the inertial manifold for the solution semigroup is acquired while operator A*satisfies the spectral interval condition.