Abstract evolution equations, periodic problems and applications.

*(English)*Zbl 0789.35001
Pitman Research Notes in Mathematics Series. 279. Harlow: Longman Scientific & Technical. New York: Wiley. iv, 249 p. (1992).

The authors present the theory of abstract semilinear parabolic equations of the form
\[
\partial_ t v(t)+ A(t)v(t)= F(t,v(t)),
\]
where the linear operator \(A(t)\) has the domain of definition independent of \(t\) and the nonlinearity is defined on an interpolation space. Special emphasize is put on time-periodic equations.

The account starts with a comprehensive treatment of linear equations. After standard results on analytic semigroups and evolution operators, properties of these operators in interpolation spaces are considered followed by a detailed discussion of linear periodic equations. Various topics of a more special nature, such as results related to maximum principles, are also included.

Regarding the nonlinear equations, included are basic results on existence, continuous dependence and extendability of solutions and a few fundamental results of the stability theory.

As applications of the abstract results, second order parabolic equations on bounded and unbounded domains are considered.

The text is self-contained to a large extend. References to results on more general abstract equations, as well as on various types of specific equations where the abstract results apply are supplied. The use of interpolation spaces, rather than fractional power spaces used in previously published books on similar topics, “makes it possible to develop a much more elegant theory of semilinear equations and is capable of extensions to problems having a much greater degree of generality…”. The book indeed provides an easy access to a big scope of results and should serve as a useful reference.

From the contents: I. Linear evolution equations of parabolic type (\(C_ 0\)-semigroups, evolution operator, interpolation spaces); II. Linear periodic equations (spectral decompositions, Floquet representations); III. Miscellaneous (Yosida approximations, parameter dependence, maximum principles and positivity); IV. Semilinear evolution equations of parabolic type (mild solutions, global solutions, parameter dependence); V. Semilinear periodic evolution equations (the period map, stability of periodic solutions); VI. Applications (reaction diffusion equations, an example in epidemiology); Appendix (function spaces, interpolation).

The account starts with a comprehensive treatment of linear equations. After standard results on analytic semigroups and evolution operators, properties of these operators in interpolation spaces are considered followed by a detailed discussion of linear periodic equations. Various topics of a more special nature, such as results related to maximum principles, are also included.

Regarding the nonlinear equations, included are basic results on existence, continuous dependence and extendability of solutions and a few fundamental results of the stability theory.

As applications of the abstract results, second order parabolic equations on bounded and unbounded domains are considered.

The text is self-contained to a large extend. References to results on more general abstract equations, as well as on various types of specific equations where the abstract results apply are supplied. The use of interpolation spaces, rather than fractional power spaces used in previously published books on similar topics, “makes it possible to develop a much more elegant theory of semilinear equations and is capable of extensions to problems having a much greater degree of generality…”. The book indeed provides an easy access to a big scope of results and should serve as a useful reference.

From the contents: I. Linear evolution equations of parabolic type (\(C_ 0\)-semigroups, evolution operator, interpolation spaces); II. Linear periodic equations (spectral decompositions, Floquet representations); III. Miscellaneous (Yosida approximations, parameter dependence, maximum principles and positivity); IV. Semilinear evolution equations of parabolic type (mild solutions, global solutions, parameter dependence); V. Semilinear periodic evolution equations (the period map, stability of periodic solutions); VI. Applications (reaction diffusion equations, an example in epidemiology); Appendix (function spaces, interpolation).

Reviewer: P.Polacik (Bratislava)

##### 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 |

35K20 | Initial-boundary value problems for second-order parabolic equations |

35B10 | Periodic solutions to PDEs |

35K55 | Nonlinear parabolic equations |

35B35 | Stability in context of PDEs |

35K57 | Reaction-diffusion equations |