Finite dimensionality and compactness of attractors for von Karman equations with nonlinear dissipation
The author proves the existence of a global, compact attractor of finite Hausdorff dimension for von Karmans shell equation with nonlinear damping: \[ w_{tt}+ \Delta^2w+ g(w_t)+ \rho\Delta w= [{\mathcal F}_f(w)+ \theta, w+f]+ F\tag{1} \] in \(Q= \Omega\times (0,\infty)\), with boundary conditions \(w= {\partial w\over\partial n}= 0\) on \(\partial\Omega\times (0,\infty)\), i.e. with \(w\in H^4(\Omega)\cap H^2_0(\Omega)\). Here \(\Delta^2{\mathcal F}_f(w)= -[w+ 2f,w]\) on \(Q\), \({\mathcal F}_f(w)\in H^4(\Omega)\cap H^2_0(\Omega)\), with \([ ,\;]\) the von Karman bracket: \([u,v]= u_{xx} v_{yy}+ u_{yy}v_{xx}- 2u_{xy} v_{xy}\). Finally, \(g: \mathbb{R}\to\mathbb{R}\) is a nonlinear damping, subject to \(0< m\leq g'(s)\), \(s\in\mathbb{R}\), and further conditions. The function \(f\in H^2_0(\Omega)\) is given and depends on the curvature of the shell, \(\theta\in H^2_0(\Omega)\cap W^{2,\infty}(\Omega)\) is the stress function due to thermal effects, while \(F\in L^2(\Omega)\) is an outer force. The proof of the existence of the global attractor turns out to be rather difficult, since many of the standard devices from attractor theory do not work in the present context, due to the nature of the nonlinearity. Theorem 1.1, proved in [\textit{A. Favini}, \textit{M. A. Horn}, \textit{I. Lasiecka} and \textit{D. Tataru}, Differ. Integral Equ. 9, No. 2, 267-294 (1996; Zbl 0847.35070)], serves as basis since it asserts global existence and uniqueness of (weak) solutions and thus provides the nonlinear semigroup \(T(t)\), \(t\geq 0\), describing the dynamics of (1). The main result is provided by Theorem 1.2, (I)--(III) which has to be summarized. Part (I) asserts the existence of an absorbing set. Part (II) states that if in addition also \[ g'(s)\leq Ms^2,\quad|s|\geq 1 \] holds, then a global compact attractor exists. Part (III) finally states that if \(g'(s)\) satisfies an additional smoothness condition then the Hausdorff dimension \(A\) is finite. The proof of Theorem 1.2 splits into several, in part rather difficult steps, contained in a series of lemmas and theorems, which are based on earlier work on the same topic by the author and coauthors.
scientific article; zbMATH DE number 1406553