Analytic semigroups and optimal regularity in parabolic problems.

*(English)*Zbl 0816.35001
Progress in Nonlinear Differential Equations and their Applications. 16. Basel: Birkhäuser. xvii, 424 p. (1995).

This book presents a systematic treatment of analytic semigroups and abstract parabolic equations. Motivated by applications to fully nonlinear problems the approach is focused on classical solutions with continuous or Hölder continuous derivatives. The generators of the analytic semigroups considered in this book are not necessarily dense defined, and hence the theory is developed without invoking results about general \(C_ 0\) semigroups.

In Chapters 0 and 1 some preliminary material is introduced: Hölder spaces, interpolation spaces, interpolatory inclusions and the reiteration theorem. Chapter 2 gives a unified development of analytic semigroups and intermediate spaces. Chapter 3 presents results about generation of analytic semigroups by second order strongly elliptic operators (with Dirichlet or first order non tangential boundary conditions) in \(L^ p\) spaces, in spaces of continuous or Hölder continuous functions, and in \(C^ 1\) spaces. Results on \(m\)-th order operators are stated without proofs. Chapter 4 concerns solvability and asymptotic behavior of the initial value problem for abstract linear nonhomogeneous equations \(u'= Au+f\). Particular attention is paid to “optimal” (or “maximal”) regularity results, that is when \(u'\) and \(Au\) enjoy the same regularity properties as \(f\). In Chapter 6 these abstract results are applied to linear parabolic initial and initial boundary value problems. Chapter 7 is devoted to (abstract and concrete) semilinear problems. Finally, the last two chapters are concerned with fully nonlinear problems (local existence and uniqueness, maximal time interval, regularity, stability of stationary solutions by linearization, bifurcation of stationary solutions into stationary or periodic solutions and their stability, invariant manifolds near stationary solutions) and their applications (mean curvature flow, detonation theory, Bellmann equations from stochastic control, free boundary value problems).

In Chapters 0 and 1 some preliminary material is introduced: Hölder spaces, interpolation spaces, interpolatory inclusions and the reiteration theorem. Chapter 2 gives a unified development of analytic semigroups and intermediate spaces. Chapter 3 presents results about generation of analytic semigroups by second order strongly elliptic operators (with Dirichlet or first order non tangential boundary conditions) in \(L^ p\) spaces, in spaces of continuous or Hölder continuous functions, and in \(C^ 1\) spaces. Results on \(m\)-th order operators are stated without proofs. Chapter 4 concerns solvability and asymptotic behavior of the initial value problem for abstract linear nonhomogeneous equations \(u'= Au+f\). Particular attention is paid to “optimal” (or “maximal”) regularity results, that is when \(u'\) and \(Au\) enjoy the same regularity properties as \(f\). In Chapter 6 these abstract results are applied to linear parabolic initial and initial boundary value problems. Chapter 7 is devoted to (abstract and concrete) semilinear problems. Finally, the last two chapters are concerned with fully nonlinear problems (local existence and uniqueness, maximal time interval, regularity, stability of stationary solutions by linearization, bifurcation of stationary solutions into stationary or periodic solutions and their stability, invariant manifolds near stationary solutions) and their applications (mean curvature flow, detonation theory, Bellmann equations from stochastic control, free boundary value problems).

Reviewer: L.Recke (Berlin)

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |

47D06 | One-parameter semigroups and linear evolution equations |

58D25 | Equations in function spaces; evolution equations |