On an extensible beam equation with nonlinear damping and source terms (Q1949005)

The paper is concerned with the global existence, stability and the longtime dynamics of solutions to the initial boundary value problem of an extensible beam equation with nonlinear damping and source terms, namely \[ u_{tt} - M(\|\nabla u\|^2)\Delta u + \Delta^2 u + g(u_t) + f (u) = h(x)\qquad \text{ in }\Omega\times \mathbb R^+, \] where \(\Omega\) is a bounded domain in \(\mathbb R^N\) with smooth boundary, either with hinged boundary condition \[ u|_{\partial\Omega}=0, \quad \Delta u|_{\partial\Omega}= 0, \] or clamped boundary condition \[ u|_{\partial\Omega} =0, \quad \frac{\partial u}{\partial \nu}\left|_{\partial\Omega}\right.= 0. \] It is proven that the problem is globally well posed, provided that either the growth exponent \(p\) of the source term \(f\) is non-supercritical, or that it is dominated by the growth exponent \(q\) of the nonlinear damping \(g\). Moreover, the related solution semigroup has a finite-dimensional global attractor and also an exponential attractor, provided either \(2\leq p\leq q\leq p^*\) or \(p = q > p^*\) if \(N\geq 5\). When the space dimension is less or equal than \(4\), then all the conclusions hold without any restrictions on \(p\) and \(q\).