\(L^{p}\)-boundedness and topological structure of solutions for flexible structural systems (Q2795277)

The paper deals with the abstract semilinear differential equation NEWLINE\[NEWLINE \alpha u'''(t) + u''(t) - \beta Au(t) - \gamma Au'(t) = f(t, u(t), u'(t)) \eqno(1) NEWLINE\]NEWLINE and its linear version \(f(t, u, u') = f(t),\) where \(A\) is a linear operator in a Banach space \(X\) and \(\alpha, \beta, \gamma\) are positive constants. The motivation is the equation describing the dynamics of flexible structures with internal material damping, where the left side of (1) is NEWLINE\[NEWLINE u'' + \lambda u''' = c^2(\Delta u + \mu \Delta u') NEWLINE\]NEWLINE with \(\Delta\) the Laplacian in a bounded domain \(\Omega\) satisfying mixed boundary conditions, \(c>0\) the wave velocity and \(0< \lambda< \mu.\) Mild solutions of (1) with given initial conditions are defined using a version of Duhamel's principle. Under suitable assumptions on the initial conditions, the operator \(A\) and the nonlinear term \(f(t, u, u')\), the authors show that the set of mild solutions of (1) in an interval \(I\) is compact and connected in \(C^1(I, X).\)