Inertial manifold for the motion of strongly damped nonlinear elastic beams (Q1386395)

The authors study an equation from elasticity of the form \[ u_{tt}+\alpha u_{xxxx}+\delta u_{xxxxt}+ \Biggl(\beta+ \int^1_0 u_x(\zeta, t)^2d\zeta\Biggr) u_{xx}= 0\tag{1} \] subject to the boundary conditions \[ u(x, 0)= u_0(x),\;u_t(x,0)= u_1(x),\;u(0, t)= u_x(0, t)= u_{xx}(1, t)= u_{xxx}(1, t)=0.\tag{2} \] They show that the initial-boundary value problem (1), (2), when cast into a proper Hilbert space setting, admits a global attractor, i.e. more explicitely, a flat inertial manifold of finite dimension. The Hilbert space setting is achieved by defining an operator \(A\) on \(H= L^2(0, 1)\) via the stipulation that \(f\in\text{dom}(A)= V\) if and only if \[ f\in H^4(0,1),\;f(0)= f'(0)= f''(1)= f'''(1)= 0. \] Operators \(A^\gamma\), \(\gamma=1/2\), \((1/2)^2\) are then defined in a standard way. In an intermediate step one rewrites (1), (2) as an abstract equation of the form: \[ u_{tt}+\alpha Au+\delta Au_t+ (\beta+\| A^{1/4}u\|^2) A^{1/2}u= 0,\;u(0)= u_0,\;u_t(0)= u_1,\tag{3} \] where \(\alpha\), \(\delta>0\), \(\beta\in\mathbb{R}\), \(u_0\in V\), \(u_1\in H\). Next one defines an operator \({\mathcal A}\) whose domain is \(V\times V\) and which assumes values in \(V\times H\) according to: \[ {\mathcal A}= \begin{pmatrix} 0 & I_V\\ -\alpha A & -\delta A\end{pmatrix},\quad I_V=\text{identity on }V.\tag{4} \] By redefining the nonlinearity in (3) suitably on \(V\times H\), (3) now assumes the form: \[ w_t={\mathcal A} w+ f(w),\quad w= (u, u_t),\quad t\geq 0.\tag{5} \] This is an evolution equation on \(E= V\times H\) since \(f\) turns out to satisfy local Lipschitz conditions. The results, mentioned at the beginning, are now proved for (5). In fact, the authors first prove the existence of an absorbing set, then of a global attractor and finally of a flat inertial manifold. A difficulty to be overcome in this connection is that the spectral gap condition does not hold. The techniques used are energy type inequalities and Gronwall inequalities.