Degenerate diffusions. Initial value problems and local regularity theory.

*(English)*Zbl 1205.35002
EMS Tracts in Mathematics 1. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-033-3/hbk). vii, 198 p. (2007).

The authors consider the degenerate and singular diffusions (even if in the title is quoted only the degenerate case). Even if this subject has been widely studied since the fifties (starting from the seminal paper by G. I. Barenblatt [Prikl. Mat. Mekh. 16, 67–78 (1952; Zbl 0049.41902)]), it still has many interesting and open questions (see for instance the monograph by E. DiBenedetto [Degenerate Parabolic Equations. Universitext. New York: Springer-Verlag (1993; Zbl 0794.35090)], which is mainly concerned with regularity issues, and the recent two by J. L. Vazquez [Smoothing and decay estimates for nonlinear diffusion equations. Equations of porous medium type. Oxford Lecture Series in Mathematics and its Applications 33. Oxford: Oxford University Press (2006; Zbl 1113.35004), The porous medium equation. Mathematical theory. Oxford Mathematical Monographs; Oxford Science Publications. Oxford: Oxford University Press (2007; Zbl 1107.35003)], where the porous medium equation is treated in full generality).

More precisely, the authors consider the equation \(u_t= \Delta\phi(u)\) where \(phi\) has either a power-like or a logarithmic behaviour and by using \(L^\infty\) estimates, Harnack-type inequalities, regularity results, they study the solvability of the Cauchy problem in the slow diffusion case (Chapter 2), for the supercritical case and for the logarithmic case (Chapter 3). In Chapter 4 they consider the case of the Cauchy-Dirichlet problem in \(D\times (0, \infty)\), where \(D\) is an open bounded set. Finally in Chapter 5 they prove that weak solutions of the porous medium equation are continuous in the degenerate case.

This book is addressed to graduate students and does an excellent job. Actually, it is a good compromise between the necessity to show the critical points of the theory and the need (in order to be readable for a student) to avoid the deep technicalities that naturally arise when one faces these questions. For these reasons, for instance, they prove only the continuity for weak solutions of degenerate porous medium equation (whereas it is well known that bounded weak solutions of degenerate (and singular too) porous medium equations are locally Hoelder continuous). Moreover, even if they essentially consider only the degenerate and supercritical case (where now the theory is quite complete thanks to the results of the last ten years), they also discuss open question especially related to the critical and subcritical case.

More precisely, the authors consider the equation \(u_t= \Delta\phi(u)\) where \(phi\) has either a power-like or a logarithmic behaviour and by using \(L^\infty\) estimates, Harnack-type inequalities, regularity results, they study the solvability of the Cauchy problem in the slow diffusion case (Chapter 2), for the supercritical case and for the logarithmic case (Chapter 3). In Chapter 4 they consider the case of the Cauchy-Dirichlet problem in \(D\times (0, \infty)\), where \(D\) is an open bounded set. Finally in Chapter 5 they prove that weak solutions of the porous medium equation are continuous in the degenerate case.

This book is addressed to graduate students and does an excellent job. Actually, it is a good compromise between the necessity to show the critical points of the theory and the need (in order to be readable for a student) to avoid the deep technicalities that naturally arise when one faces these questions. For these reasons, for instance, they prove only the continuity for weak solutions of degenerate porous medium equation (whereas it is well known that bounded weak solutions of degenerate (and singular too) porous medium equations are locally Hoelder continuous). Moreover, even if they essentially consider only the degenerate and supercritical case (where now the theory is quite complete thanks to the results of the last ten years), they also discuss open question especially related to the critical and subcritical case.

Reviewer: Vincenzo Vespri (Firenze)

##### MSC:

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

35K55 | Nonlinear parabolic equations |

35K65 | Degenerate parabolic equations |