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

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

**Introduction**- 0. General notation
**I. Linear evolution equations of parabolic type**- 1.
*C*-semigroups_{0} - 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
**R**^{n} - 26. A nonstandard example arising in epidemiology
**Appendix**- A1. Spaces of continuous and differentiable functions
- A2. Distributions and test functions
- A3. Sobolev spaces and interpolation
- A4. Boundary spaces and the trace operator
**Bibliography**