The present monograph is devoted to the theory of general parabolic boundary value problems. The vastness of this theory forced us to take difficult decisions in selecting the results to be presented and in determining the degree of detail needed to describe their proofs. In the first chapter we define the basic notions at the origin of the theory of parabolic boundary value problems and give various examples of illustrative and descriptive character. The main part of the monograph (Chapters II to V) is devoted to a the detailed and systematic exposition of the L -theory of parabolic 2 boundary value problems with smooth coefficients in Hilbert spaces of smooth functions and distributions of arbitrary finite order and with some natural appli cations of the theory. Wishing to make the monograph more informative, we included in Chapter VI a survey of results in the theory of the Cauchy problem and boundary value problems in the traditional spaces of smooth functions. We give no proofs; rather, we attempt to compare different results and techniques. Special attention is paid to a detailed analysis of examples illustrating and complementing the results for mulated. The chapter is written in such a way that the reader interested only in the results of the classical theory of the Cauchy problem and boundary value problems may concentrate on it alone, skipping the previous chapters.

Inhalt
I Equations and Problems.- I.1 Equations.- I.2 Initial and boundary value problems.- II Functional Spaces.- II.1 Spaces of test functions and distributions.- II.2 The Hilbert spaces Hs and ?s.- II.3 Banach spaces of Hölder functions.- III Linear Operators.- III.1 Operators of potential type.- III.2 Operators of multiplication by a function.- III.3 Commutators. Green formulas.- III.4 On equivalent norms in ?s(?+n+1,?), ?s(E+n+1,?), and Hs(?+n), s ? 0.- III.5 The spaces $${{\tilde{H}}^{s}}$$ and $${{\tilde{\mathcal{H}}}^{s}}$$.- III.6 Differential operators in the space $${{\tilde{\mathcal{H}}}^{s}}$$.- IV Parabolic Boundary Value Problems in Half-Space.- IV.1 Non-homogeneous systems in the space ?++s(?n+1,?).- IV.2 Initial value and Cauchy problems for parabolic systems in spaces ?s.- IV.3 Model parabolic boundary value problems in $$\bar{\mathbb{R}}{{ + + }}^{{n + 1}}$$.- IV.4 The model boundary value problemin in $$\bar{\mathbb{R}}{{ + + }}^{{n + 1}}$$ for general parabolic systems.- IV.5 The model parabolic conjugation problem in classes of smooth functions.- IV.6 Boundary value problem in $$\tilde{\mathcal{H}}{ + }^{s}(\bar{\mathbb{R}}{{ + + }}^{{n + 1}},\gamma )$$ for operators in which the coefficients of the highest-order derivatives are slowly varying functions.- IV.7 Conjugation problem for operators in which the coefficients of the highest-order derivatives are slowly varying.- V Parabolic Boundary Value Problems in Cylindrical Domains.- V.1 Boundary value problems in a semi-infinite cylinder.- V.2 Nonlocal boundary value problems. Conjugation problems.- V.3 Boundary value problems in cylindrical domains of finite height.- V.4 Solvability of the parabolic boundary value problems for right-hand sides with regular singularities.- V.5 Greenformula, boundary and initial values of weak generalized solutions.- VI The Cauchy Problem and Parabolic Boundary Value Problems in Spaces of Smooth Functions.- VI.1 Fundamental solutions of the Cauchy problem.- VI.2 The Cauchy problem.- VI.3 Schauder theory of parabolic boundary value problems.- VI.4 Green functions.- VII Behaviour of Solutions of Parabolic Boundary Value Problems for Large Values of Time.- VII.1 Asymptotic representations and stabilization of solutions of model problems.- VII.2 Tikhonov's problem.- Comments.- References.