Attractors for a class of extensible beam equation with semi-local damping

This paper is concerned with the well-posedness and long-time dynamics of a class of extensible beam equations with a semi-local nonlinear damping term given by $$-\gamma\mbox{div}\left[g(\nabla u_t)\right],\,\,(\gamma>0),$$ where $g\in C^1(\mathbb{R}^n,\mathbb{R}^n)$ is a gradient field of the form $g(s_1,\cdots,s_n)=(g_1(s_1),\cdots,g_n(s_n))$ with $g_i$ a monotone increasing function such that $g_i(0)=0$ and satisfies the growth condition $k|s|^{\alpha}\le g’_i(s)\le K(1+|s|^{\alpha})$ for all $s\in \mathbb{R}$, $i=1,\cdots,n$ and for some positive constants $k,K$ and $\alpha\ge 0$. This kind of damping arise in connection to the (fundamentally unbounded) nonlinearity proposed by Prestel in \cite{Prestel} in the context of wave equation. Our main results concern decay rates, existence of attractors and their properties.