Abstract evolution equations, periodic problems and applications

Daniel Daners and Pablo Koch Medina
Pitman Research Notes in Mathematics Series 279
Longman Scientific & Technical, 1992
PDF version of the book (identical to original except for correction of minor typos)
Citations on Google Scholar


The book is an introduction to abstract semi-linear evolution equations of parabolic type, with special emphasis on periodic problems. Throughout, it makes use of the theory of interpolation spaces rather than fractional power spaces. While the use of fractional power spaces involves a lot of technicalities, the use of interpolation spaces allows a much more elegant and complete treatment of semi-linear problems, particularly for non-autonomous equations. At the same time it brings more conceptual clarity. It is then shown how these abstract results can be applied to concrete reaction-diffusion equations and systems. The table of contents is given below.

Many of the results appear for the first time in book form and thus, these notes should serve as a useful reference.

Readership: Researchers and postgraduate students in linear and nonlinear differential equations and dynamical systems.

An errata for the original version is available: dvi-file (5.7kB), ps-file (38kB), PDF-file (70kB)

If you find more errors or have other comments e-mail me.

AMS Subject Classification (1991): 35Axx, 35Bxx, 35Kxx


0. General notation
I. Linear evolution equations of parabolic type
1. C0-semigroups
2. The evolution operator
3. Interpolation spaces
4. The real, complex and continuous interpolation methods
5. The evolution operator in interpolation spaces
II. Linear periodic evolution equations
6. The evolution operator
7. Spectral decompositions
8. Floquet representations
III. Miscellaneous
9. Abstract Volterra integral equations
10. Yosida approximations of the evolution operator
11. Parameter dependence
12. Ordered Banach spaces and positive operators
13. The parabolic maximum principle and positivity
14. Superconvexity and periodic-parabolic eigenvalue problems
IV. Semilinear evolution equations of parabolic type
15. Mild solutions
16. Existence and continuous dependence
17. Global solutions
18. Parameter dependence
V. Semilinear periodic evolution equations
19. Equilibria in autonomous equations
20. The period-map
21. Stability of periodic solutions
22. Linearized stability and instability
23. Stability in weaker norms
VI. Applications
24. Reaction-diffusion equations in bounded domains
25. Reaction-diffusion equations in Rn
26. A nonstandard example arising in epidemiology
A1. Spaces of continuous and differentiable functions
A2. Distributions and test functions
A3. Sobolev spaces and interpolation
A4. Boundary spaces and the trace operator