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 +(-?)?ut + f(u)= g(x),with a ?[1,2)and nonlinearity f(u)with the growth exponent p.Especially,when ? = 1,p*= pa = p?'.When 1 ? p<p*(=N+4/(N-4)+)and p*? p<p?(==N+4?/(N-4?)+)),in the case of non-supercitical and supercritical condition-s,we prove the existence of the solutions.When 1 ? p<p?,the solutions of the wave equation is stable in Y? = V? × V-?.When 1<p<pa,the solutions of the wave equa-tions are of higher regularity in X? = V?+1 × V?.When 1<p<pa,the related solution semigroup has a finite fractal dimensional global attractor with strong topology.When 1 ? p<p?'(= N+2(?+1)/(N-2(?+1))+),the related solution semigroup has an exponential attractor with strong topology.When p?'?p<P?,the related solution semigroup has an exponential attractor with partially strong topology.Finally,we prove the upper semi-continuity of the global attractor.