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An inertial manifold for time-discretization with a nonselfadjoint operator. (Chinese. English summary) Zbl 1088.37515

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 [P. Constantin et al., Integral manifolds and inertial manifolds for dissipative partial differential equations, Springer, New York (1989; Zbl 0683.58002); C. Foias, G. R. Sell and 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.

MSC:

37L25 Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems
34G20 Nonlinear differential equations in abstract spaces
35B42 Inertial manifolds
37L65 Special approximation methods (nonlinear Galerkin, etc.) for infinite-dimensional dissipative dynamical systems
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