Analytic Semigroups and Optimal Regularity in Parabolic Problems

This book gives a systematic treatment of the basic theory of analytic semigroups and abstract parabolic equations in general Banach spaces, and of how such a theory may be used in parabolic PDE's. It takes into account the developments of the theory during the last fifteen years, and it is focused on classical solutions, with continuous or Holder continuous derivatives. On one hand, working in spaces of continuous functions rather than in Lebesgue spaces seems to be appropriate in view of the number of parabolic problems arising in applied mathematics, where continuity has physical meaning; on the other hand it allows one to consider any type of nonlinearities (even of nonlocal type), even involving the highest order derivatives of the solution, avoiding the limitations on the growth of the nonlinear terms required by the LP approach. Moreover, the continuous space theory is, at present, sufficiently well established. For the Hilbert space approach we refer to J. L. LIONS - E. MAGENES [128], M. S. AGRANOVICH - M. l. VISHIK [14], and for the LP approach to V. A. SOLONNIKOV [184], P. GRISVARD [94], G. DI BLASIO [72], G. DORE - A. VENNI [76] and the subsequent papers [90], [169], [170]. Many books about abstract evolution equations and semigroups contain some chapters on analytic semigroups. See, e. g. , E. HILLE - R. S. PHILLIPS [100]' S. G. KREIN [114], K. YOSIDA [213], A. PAZY [166], H. TANABE [193], PH.

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.