Attractors, shadowing, and approximation of abstract semilinear differential equations

S. I. Piskarev ; A. V. Ovchinnikov

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Functional Analysis, Tome 189 (2021), pp. 3-130.

Résumé

The review covers such sections of the theory of approximation of abstract differential equations as the approximation of attractors in the case of hyperbolic stationary points, the shadowing and, finally, the approximation of fractional in time semilinear problems.

Détail

Mots-clés : abstract parabolic equations
Keywords: general approximation scheme, compact convergence of resolvents, attractor, unstable manifold, stable manifold, upper and lower semicontinuity of attractors, affinity principle, principle of compact approximation, semilinear differential equation, Banach space, periodic solution, Lyapunov stability, hyperbolic equilibrium, semiflow, rotation of a vector field, index of a solution, shadowing, analytic $C_0$-semigroup, semidiscretization, discretization in space, discretization in time, fractional equation, fractional power of an operator, condensing operator.

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