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.- 1.3 Bibliographical remarks.- 2 Analytic semigroups and intermediate spaces.- 2.1 Basic properties of etA.- 2.2 Intermediate spaces.- 2.3 Spectral properties and asymptotic behavior.- 2.4 Perturbations of generators.- 2.5 Bibliographical remarks.- 3 Generation of analytic semigroups by elliptic operators.- 3.1 Second order operators.- 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.- 4.4 The nonhomogeneous problem in unbounded time intervals.- 4.5 Bibliographical remarks.- 5 Linear parabolic problems.- 5.1 Second order equations.- 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.- 6.4 Bibliographical remarks.- 7 Semilinear equations.- 7.1 Local existence and regularity.- 7.2 A priori estimates and existence in the large.- 7.3 Some examples.- 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.- 8.4 The case where X is an interpolation space.- 8.5 Examples and applications.- 8.6 Bibliographical remarks.- 9 Asymptotic behavior in fully nonlinear equations.- 9.1 Behavior near stationary solutions.- 9.2 Critical casesof stability.- 9.3 Periodic solutions.- 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.