Finite dimensional attractor for a class of extensible beams with nonlocal strong damping

This paper is concerned with the well-posedness and long-time behavior of solutions for the following extensible beams equation with the nonlocal strong damping utt+Δ2u−m(‖∇u‖2)Δu−(∇utp+1)Δut+f(u)=h. More specifically, under some suitable conditions on the Kirchhoff coefficient m(‖∇u‖2) and the nonlinear term f(u), we establish the well-posedness by means of the monotone operator theory with locally Lipschitz perturbation. And we show that the related solution semigroup {St}t≥0 in phase space H has a finite-dimensional global attractor A which has some regularity when the growth exponent of the nonlinearity f(u) is up to the subcritical and critical case, respectively.