Haraux, Alain Sharp estimates of bounded solutions to a second-order forced equation with structural damping. (English) Zbl 1184.34066 Differ. Equ. Appl. 1, No. 3, Article ID 18, 341-347 (2009). The author investigates the equation \[ u''+cAu'+Au^2=f(t), \]where \(A:H\supset D(A)\to H\) is a possibly unbounded self-adjoint linear operator on a real Hilbert space \(H\), such that \(\forall u\in D(A)\) \((Au|u)\geq|u|^2\). Here \((\cdot|\cdot)\) and \(|\cdot|\) denote the inner product and the norm in \(H\), respectively, \(c\) is a positive constant and \(f\in L^\infty(\mathbb{R},H)\). The main result states that the unique bounded solution to the equation satisfies the estimation \(\forall t\in \mathbb{R}\) \(\|u(t)\|\leq\frac{\max\{1,\frac 2c\}}{\lambda_1} \|f\|_{L^\infty(\mathbb{R},H)}\), where \(\lambda_1=\inf_{u\in D(A),|u|=1} (Au|u)\), \(\|v\|:=|Av|\) for \(v\in D(A)\). An application to the size of attractors of some nonlinear plate equation in a bounded domain is given. Reviewer: S. Burys (Kraków) Cited in 1 Document MSC: 34G10 Linear differential equations in abstract spaces 34C11 Growth and boundedness of solutions to ordinary differential equations 34D45 Attractors of solutions to ordinary differential equations Keywords:second-order equation; bounded solution; structural damping PDFBibTeX XMLCite \textit{A. Haraux}, Differ. Equ. Appl. 1, No. 3, Article ID 18, 341--347 (2009; Zbl 1184.34066) Full Text: Link