In this paper, we establish some new results on the dynamics of a class of nonlinear beam equations with degenerate energy structural damping (the Balakrishnan–Taylor (B–T) model) on a bounded domain \(\Omega \subset \mathbb {R}^N (N\ge 1){:}\,u_{tt}+\Delta ^2 u-\kappa \Delta u -\Vert U(t)\Vert _{\mathcal {H}}^{2q}\Delta u_t+f(u)=0\) , with \(\kappa \ge 0, q>0\) and either the hinged or the clamped boundary condition. The main results are as follows: (i) the existence, uniqueness and polynomial decay of the weak solutions in natural energy space \(\mathcal {H}\) provided that the nonlinearity f(u) is of optimal polynomial growth \(p{:}\, 1\le p<\frac{N+4}{N-4}\) , with \(N\ge 5\) ; (ii) the additional regularity which guarantees that the weak solutions are exactly the strong ones, and the existence of the global attractor provided that \(1\le p\le \frac{N}{N-4}\) ; (iii) especially when \(q\ge 1/2, 1\le p\le \frac{N}{N-4}\) , the weak solutions are of higher regularity when \(t>0\) , and the associated dynamical system has a strong \((\mathcal {H},\mathcal {H}_2)\) -global attractor, which is also the standard one, where \(\mathcal {H}_2\) is the strong solution space. The most distinct aspect of this paper are that: (a) it removes the longstanding restriction \(q\ge 1\) in the literature before; (b) above mentioned results not only fill the gap left open for the B–T model with the clamped boundary condition, but also improve and deepen greatly the results for the B–T model with the hinged boundary condition in recent literature. The methodology developed here allows overcoming the degeneration of the energy structural damping and realizing the purpose.
