An inertial manifold for time-discretization with a nonselfadjoint operator. (Q2767347)

In this paper the existence of inertial manifolds of the form \(M_h=\text{Graph}(\phi_h)\), where \(\phi_h\) is the fixed point of the inertial map, is proved under the assumption that \(h\) is small enough and the spectral gap conditions are satisfied for the time-discrete equation \((u^{n+1}-u^n)/h+Au^{n+1}=F(u^n)\). Our paper is different from the other works [\textit{P. Constantin} et al., Integral manifolds and inertial manifolds for dissipative partial differential equations, Springer, New York (1989; Zbl 0683.58002); \textit{C. Foias, G. R. Sell} and \textit{R. Temam}, J. Differ. Equations 73, No. 2, 309--353 (1988; Zbl 0643.58004)] in that the operator \(A\) in this paper is only an infinitesimal generator of an analytic semigroup and has compact resolvent, but is not assumed to be selfadjoint.