Generalized convergence and uniform bounds for semigroups of restrictions of nonselfadjoint operators (Q607510)

In [Nonlinear Anal., Theory Methods Appl. 69, No.~9 (A), 3110--3127 (2008; Zbl 1149.35016)], the author showed the existence of exponentially attracting invariant manifolds for a nonlinear strongly damped viscoelastic problem of the type \[ \left\{\begin{aligned} &u_{tt}(x,t)- u_{xx}(x,t)-\alpha u_{txx}(x,t)+\varepsilon f\biggl(u(1,t), \frac{u_t(1,t)}{\sqrt{\varepsilon}}\biggr)=0,\;0<x<1,\;t>0,\\ &u(0,t)=0,\\ &u_{tt}(1,t)= -\varepsilon[u_x+\alpha u_{tx}+ ru_t](1,t)-\varepsilon f\biggl(u(1,t), \frac{u_t(1,t)}{\sqrt{\varepsilon}}\biggr), \end{aligned}\right. \] where \(\alpha, r>0\) and \(\varepsilon\geq 0.\) By using the notion of generalized convergence between operators, the author proves in the paper under review a special result (used and not proved in [op.\,cit.]), concerning certain uniform bounds for families of semigroups generated by restrictions of nonselfadjoint linear operators.