Longtime Dynamics Of A Class Of Extensible Beam Equations With Structural Damping

The paper is concerned with longtime dynamics of a class of extensible beam equations:utt-M(||▽u||2)△u + △2u +(-△)αut + f(u)= g(x),with a ∈[1,2)and nonlinearity f(u)with the growth exponent p.Especially,when α = 1,p*= pa = pα’.When 1 ≤ p<p*(=N+4/(N-4)+)and p*≤ p<pα(==N+4α/(N-4α)+)),in the case of non-supercitical and supercritical condition-s,we prove the existence of the solutions.When 1 ≤ p<pα,the solutions of the wave equation is stable in Yα = Vα × V-α.When 1<p<pa,the solutions of the wave equa-tions are of higher regularity in Xα = Vα+1 × Vα.When 1<p<pa,the related solution semigroup has a finite fractal dimensional global attractor with strong topology.When 1 ≤ p<pα’(= N+2(α+1)/(N-2(α+1))+),the related solution semigroup has an exponential attractor with strong topology.When pα’≤p<Pα,the related solution semigroup has an exponential attractor with partially strong topology.Finally,we prove the upper semi-continuity of the global attractor.