The well-posedness and attractor on an extensible beam equation with nonlocal weak damping

In this paper, we consider some new results on the well-posedness and the asymptotic behavior of the solutions for a class of extensible beams equation with the nonlocal weak damping and nonlinear source terms. Our contribution is threefold. First, we establish the well-posedness by means of the monotone operator theory with locally Lipschitz perturbation. Then we show that the related solution semigroup $ \{S_{t}\}_{t\geq0} $ in phase space $ \mathcal{H} $ has a finite-dimensional global attractor $ \mathcal{A} $ which has some regularity when the growth exponent of the nonlinearity $ f(u) $ is up to the subcritical and critical case, respectively. Finally, we obtain the exponential attractor $ \mathcal{A}_{exp} $ of the dynamical system $ (\mathcal{H}, S_{t}) $. These results deepen and extend our previous works([31], [30]), where we only considered the existence of the global attractors in the case of degenerate damping.