Longtime Dynamics For The Kirchhoff Type Wave Equation With Structural Damping

The thesis studies longtime dynamics of the Kirchhoff type equation with structural damping: utt-M(‖▽u‖2)Δu+(--Δ)αut+f(u)=g(x). The main results are concerned with fractional damping (-Δ)αut with α∈(1/2,1) and nonlinearity f(u) with supercrit-ical growth exponent. When 1 ≤p <(N+4a/(N-4a), we prove that the solution for the equations is of higher global regularity (not partially regularity as usual) and the re-late solution semigroup has global and exponential attractors in natural energy space [H01(Ω)∩Lp+1(Ω]×L2(Ω).The chapter five is concerned with the well-posedness of the Kirchhoff type equation with structural damping: utt+(-Δ)αut-M(‖▽u‖)2Δu+ut+g(x,u)=f(x) in RN ×R+ u,(x,0)=u0(x),ut(x,0)=u,1(x),x∈RN N≥3, M(s),g(x,u) is nonlinearity ,f(x) is external force. We prove the well-posedness N+2c~and regularity of the solution as .