Longtime Dynamics Of The Kirchhoff Type Wave Equations With Gentle Damping

The paper is concerned with longtime dynamics of the Kirchhoff type wave equations:u_tt-M（||▽u||²）△u+▽2u+（-Δ）αut+f（u） =g（x）, with α∈（0,1） and nonlinearity f（u）with the growth exponent p. As 1≤p<pα≡N+4α/（N-4）⁺,the solutions of the wave equations is of higher global regularity （not partially regularity as usual） and the relate solution semigroup has a finite fractal dimensional global attractor and an exponential attractor in natural energy space [H²（Ω）∩H₀¹（Ω）]×L²（Ω）. As p_α≤p<p*≡N+4/N-4, the subclass G of limit solutions has a weak global attractor.