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**Linear and quasilinear parabolic problems. Vol. 1: Abstract linear theory.**
*(English)*
Zbl 0819.35001

Monographs in Mathematics. 89. Basel: Birkhäuser. xxxv, 335 p. (1995).

This monograph is the first of three volumes which present the semigroup approach to abstract quasilinear evolution equations of parabolic type that has been developed largely by the author over the last ten years. This approach is, on the one hand, flexible, and, on the other hand, general enough to cover a remarkable wide variety of (possibly nonlinear dynamic) boundary value problems for (possibly noncoercive) quasilinear parabolic systems in an \(L_ p\)-setting.

The first volume concerns abstract linear parabolic evolution equations in general Banach spaces. The second volume (“Function spaces and linear differential operators”) is devoted to concrete realizations of these equations by linear parabolic systems, and the last volume (“Nonlinear problems”) contains the nonlinear theory.

This first volume is organized into five chapters. Chapter I (“Generators and interpolation”) presents the generators of analytic semigroups and the interpolation functors that are used throughout. Chapter II (“Cauchy problems and evolution operators”) deals with abstract nonautonomous linear parabolic Cauchy problems, with the corresponding evolution operators and with their properties (stability estimates, positivity properties in the case of ordered Banach spaces). Chapter III (“Maximal regularity”) presents abstract linear parabolic evolution equations such that the derivatives of the solutions are as regular as the right-hand sides of the equations. Results on maximal regularity are obtained in a Sobolev space setting and in spaces of Hölder continuous resp. of uniformly continuous functions with prescribed singularity at zero. Chapter IV (“Variable domains”) concerns equations \(\dot u+ A(t)u= f(t)\) where it is not assumed that an interpolation space between the basic space and the domain of definition of \(A(t)\) is independent of \(t\). The results are applied to abstract initial boundary value problems. Finally, in chapter V (“Scales of Banach spaces”) linear parabolic evolution equations in continuous scales of Banach spaces are considered. It is shown that the theory of interpolation-extrapolation scales can be used to prove that the solutions of the “extrapolated weak” equations are, in fact, solutions of the original problem. Moreover, the extrapolation spaces often allow the reduction of problems with variable domains to fixed domain problems.

The first volume concerns abstract linear parabolic evolution equations in general Banach spaces. The second volume (“Function spaces and linear differential operators”) is devoted to concrete realizations of these equations by linear parabolic systems, and the last volume (“Nonlinear problems”) contains the nonlinear theory.

This first volume is organized into five chapters. Chapter I (“Generators and interpolation”) presents the generators of analytic semigroups and the interpolation functors that are used throughout. Chapter II (“Cauchy problems and evolution operators”) deals with abstract nonautonomous linear parabolic Cauchy problems, with the corresponding evolution operators and with their properties (stability estimates, positivity properties in the case of ordered Banach spaces). Chapter III (“Maximal regularity”) presents abstract linear parabolic evolution equations such that the derivatives of the solutions are as regular as the right-hand sides of the equations. Results on maximal regularity are obtained in a Sobolev space setting and in spaces of Hölder continuous resp. of uniformly continuous functions with prescribed singularity at zero. Chapter IV (“Variable domains”) concerns equations \(\dot u+ A(t)u= f(t)\) where it is not assumed that an interpolation space between the basic space and the domain of definition of \(A(t)\) is independent of \(t\). The results are applied to abstract initial boundary value problems. Finally, in chapter V (“Scales of Banach spaces”) linear parabolic evolution equations in continuous scales of Banach spaces are considered. It is shown that the theory of interpolation-extrapolation scales can be used to prove that the solutions of the “extrapolated weak” equations are, in fact, solutions of the original problem. Moreover, the extrapolation spaces often allow the reduction of problems with variable domains to fixed domain problems.

Reviewer: L.Recke (Berlin)

### MSC:

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

35G10 | Initial value problems for linear higher-order PDEs |

35K25 | Higher-order parabolic equations |

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

58D25 | Equations in function spaces; evolution equations |