Analytic Semigroups and Optimal Regularity in Parabolic Problems

ISBN-13: 9783034899567 / Angielski / Miękka / 2011 / 424 str.

Alessandra Lunardi

Opis

The book shows how the abstract methods of analytic semigroups and evolution equations in Banach spaces can be fruitfully applied to the study of parabolic problems.
Particular attention is paid to optimal regularity results in linear equations. Furthermore, these results are used to study several other problems, especially fully nonlinear ones.
Owing to the new unified approach chosen, known theorems are presented from a novel perspective and new results are derived.
The book is self-contained. It is addressed to PhD students and researchers interested in abstract evolution equations and in parabolic partial differential equations and systems. It gives a comprehensive overview on the present state of the art in the field, teaching at the same time how to exploit its basic techniques.

Spis treści

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.

O autorze

Alessandra Lunardi is a professor of mathematics at the University of Parma, Italy.