Attractors and approximate inertial manifolds for the generalized Benjamin-Bona-Mahony equation (Q2785657)

The author investigates a generalized Benjamin-Bona-Mahony equation, i.e. NEWLINE\[NEWLINEu_t- u_{xxt}- \nu u_{xx}+ f(u)_x= g(x)\quad\text{on }\Omega\times \mathbb{R}_+\tag{1}NEWLINE\]NEWLINE from the point of view of dynamical systems. Here \(\Omega\) is a bounded interval, \(\nu>0\), \(f\in C^\infty(\mathbb{R},\mathbb{R})\) and \(g(x)\in L^2(\Omega)\) subject to \(\int_\Omega g(x)dx=0\). In the introduction, the author discusses the work of previous authors related to the case \(f(u)= u\) and \(g=0\). The aim of the paper is to prove the existence of a finite, global weak attractor for (1) and of approximate inertial manifolds.NEWLINENEWLINENEWLINESection 2 is devoted to the proof of a series of estimates and inequalities of technical nature. In section 3 the first main result is proved. To this effect, sets \(O_k= \{u\in H^k/|u|_{H'}\leq E_k\}\) are defined where \(H^k= H^k(\Omega)\) and \(E_k\), \(k\geq 0\) are constants depending only on the given data.NEWLINENEWLINENEWLINEWith \(S(t)\), \(t\geq 0\) the global semiflow generated by (1) one defines sets \(A_k=\bigcap_{k\geq 2}\) closure \((\bigcup_{t\geq 0} S(t)O_k)\) for \(k\geq 2\). Theorem 3.1 then states that if \(g\in H^{k-1}\), \(k\geq 2\) then \(A_k\) has the properties: (i) \(A_k\) is bounded and weakly closed in \(H^k\cap H_\alpha\), (ii) \(S(t)A_k= A_k\), \(t\geq 0\), (iii) for every bounded set \(B_k\) in \(H^k\), \(S(t)B_k\) converges to \(A_k\) with respect to the \(H^k\)-weak topology as \(t\to\infty\). In (i), \(H_\alpha\) is an auxiliary set defined as follows: \(H_\alpha=\{u\in H^0/|\theta(u)|\leq \alpha\}\) while \(\theta(u)=|\Omega|^{-1}\int_\Omega u(x)dx\). Subsequent to the proof of Theorem 3.1 a technically defined integer \(m\geq 1\) is introduced whose properties are described by Theorem 3.2 which says that the Hausdorff dimension of \(A_k\) is \(\leq m\) while its fractal dimension is \(\leq 2m\). The last section is devoted to the proof of the existence of so-called approximate inertial manifolds.