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Sharp estimates of bounded solutions to a second-order forced equation with structural damping. (English) Zbl 1184.34066

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)

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
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