Asymptotic Behavior Of The Kirchhoff Type Equation

The paper studies the long time behavior of the Kirchhoff type equation with strongdampingwhere M(s)=1+s^m/2,m≥1.Ω(?)R^N is a bounded domain with smooth boundary (?)Ω.It proves that the corresponding IBVP possesses an unique solution both locally andglobally in time, u∈C([0,+∞);H²âˆ©H₀¹)∩C¹([0,+∞);H₀¹). We define a mapping S(t) :Xâ†'X, where X=H²âˆ©H₀¹Ã—H₀¹,S(t)(u₀,u₁)=(u,u_t), then S(t) is a Co-semigroupin X. It proves that continuous semigroup S(t) has an absorbing set in X. With twodifferent methords, it proves that the continuous semigroup S(t) has a global attractor inphase space X. At last, we give some examples to show the existence of the nonlinearfunctions g(x,u) and h(u_t).