Longtime Behavior Of The Non-degenerate Kirchhoff Type Equation With Strong Damping

In this paper, We consider the initial boundary value problem of the non-degenerate Kirchhoff equation with strong damping:whereΩ(?) Rⁿ is a bounded domain with smooth boundary (?); g(s) and h(s) are nonlinear functions, f(x) is an external force term.The problem (0.1) is equivalent to the following problem (for brevity, we takeα=1 in the sequel):where A = -Δand D(A) = H²(Ω)∩H₀¹(Ω). Since A is self-adjoint and strictly positive on V₁ = H₀¹(Ω), we can define the powers A^s of A(s∈R) and the Hilbert spaces V_s = D(A^?)(s > 0), where V₁ =H₀¹(Ω), V₂ = H²(Ω)∩H₀¹(Ω), H = L²(Ω).We prove that the problem (0.2) exists an unique global solution in the phase space V₁×H by Galerkin method, and the related continuous semigroup has absorbing set in V₁×H. Furthermore, we prove that the problem (0.2) exists an unique solution in V_1+δ×V_δ, and the continuous semigroup has absorbing set in V_1+δ×V_δ. We get that there exists a global attractor of dynamical system (0.2) in V_1+δ×V_δby proving the asymptotic compactness of the related continuous semigroup.