Analytic Semigroups and Optimal Regularity in Parabolic Problems

0 Preliminary material: spaces of continuous and Hölder continuous functions.- 0.1 Spaces of bounded and/or continuous functions.- 0.2 Spaces of Hölder continuous functions.- 0.3 Extension operators.- 1 Interpolation theory.- 1.1 Interpolatory inclusions.- 1.2 Interpolation spaces.- 1.2.1 The K-method.- 1.2.2 The trace method.- 1.2.3 The Reiteration Theorem.- 1.2.4 Some examples.- 1.3 Bibliographical remarks.- 2 Analytic semigroups and intermediate spaces.- 2.1 Basic properties of etA.- 2.1.1 Identification of the generator.- 2.1.2 A sufficient condition to be a sectorial operator.- 2.2 Intermediate spaces.- 2.2.1 The spaces DA(?, p) and DA(?).- 2.2.2 The domains of fractional powers of —A.- 2.3 Spectral properties and asymptotic behavior.- 2.3.1 Estimates for large t.- 2.3.2 Spectral properties of etA.- 2.4 Perturbations of generators.- 2.5 Bibliographical remarks.- 3 Generation of analytic semigroups by elliptic operators.- 3.1 Second order operators.- 3.1.1 Generation in Lp(?), 1 < p < ?.- 3.1.2 Generation in L? (Rn) and in spaces of continuous functions in Rn.- 3.1.3 Characterization of interpolation spaces and generation results in Hölder spaces in Rn.- 3.1.4 Generation in C1(Rn).- 3.1.5 Generation in L? (?) and in spaces of continuous functions in $$ \overline \Omega $$.- 3.2 Higher order operators and bibliographical remarks.- 4 Nonhomogeneous equations.- 4.1 Solutions of linear problems.- 4.2 Mild solutions.- 4.3 Strict and classical solutions, and optimal regularity.- 4.3.1 Time regularity.- 4.3.2 Space regularity.- 4.3.3 A further regularity result.- 4.4 The nonhomogeneous problem in unbounded time intervals.- 4.4.1 Bounded solutions in [0, +?[.- 4.4.2 Bounded solutions in ] - ?, 0].- 4.4.3 Bounded solutions in R.- 4.4.4 Exponentially decaying and exponentially growing solutions.- 4.5 Bibliographical remarks.- 5 Linear parabolic problems.- 5.1 Second order equations.- 5.1.1 Initial value problems in [0,T] × Rn.- 5.1.2 Initial boundary value problems in $$ \left[ {0,T} \right] \times \overline \Omega $$.- 5.2 Bibliographical remarks.- 6 Linear nonautonomous equations.- 6.1 Construction and properties of the evolution operator.- 6.2 The variation of constants formula.- 6.3 Asymptotic behavior in the periodic case.- 6.3.1 The period map.- 6.3.2 Estimates on the evolution operator.- 6.3.3 Asymptotic behavior in nonhomogeneous problems.- 6.4 Bibliographical remarks.- 7 Semilinear equations.- 7.1 Local existence and regularity.- 7.1.1 Local existence results.- 7.1.2 The maximally defined solution.- 7.1.3 Further regularity, classical and strict solutions.- 7.2 A priori estimates and existence in the large.- 7.3 Some examples.- 7.3.1 Reaction-diffusion systems.- 7.3.2 A general semilinear equation.- 7.3.3 Second order equations with nonlinearities in divergence form.- 7.3.4 The Cahn-Hilliard equation.- 7.4 Bibliographical remarks for Chapter 7.- 8 Fully nonlinear equations.- 8.1 Local existence, uniqueness and regularity.- 8.2 The maximally defined solution.- 8.3 Further regularity properties and dependence on the data.- 8.3.1 Ck regularity with respect to (x, ?).- 8.3.2 Ck regularity with respect to time.- 8.3.3 Analyticity.- 8.4 The case where X is an interpolation space.- 8.5 Examples and applications.- 8.5.1 An equation from detonation theory.- 8.5.2 An example of existence in the large.- 8.5.3 A general second order problem.- 8.5.4 Motion of hypersurfaces by mean curvature.- 8.5.5 Bellman equations.- 8.6 Bibliographical remarks.- 9 Asymptotic behavior in fully nonlinear equations.- 9.1 Behavior near stationary solutions.- 9.1.1 Stability and instability by linearization.- 9.1.2 The saddle point property.- 9.1.3 The case where X is an interpolation space.- 9.1.4 Bifurcation of stationary solutions.- 9.1.5 Applications to nonlinear parabolic problems, I.- 9.1.6 Stability of travelling waves in two-phase free boundary problems.- 9.2 Critical cases of stability.- 9.2.1 The center-unstable manifold.- 9.2.2 Applications to nonlinear parabolic problems, II.- 9.2.3 The case where the linear part generates a bounded semigroup.- 9.2.4 Applications to nonlinear parabolic problems, III.- 9.3 Periodic solutions.- 9.3.1 Hopf bifurcation.- 9.3.2 Stability of periodic solutions.- 9.3.3 Applications to nonlinear parabolic problems, IV.- 9.4 Bibliographical remarks.- Appendix: Spectrum and resolvent.- A.1 Spectral sets and projections.- A.2 Isolated points of the spectrum.- A.3 Perturbation results.