**Mathematical Surveys and Monographs**

Volume: 192;
2013;
221 pp;
Hardcover

MSC: Primary 35; 37;

**Print ISBN: 978-1-4704-0985-2
Product Code: SURV/192**

List Price: $101.00

AMS Member Price: $80.80

MAA Member Price: $90.90

**Electronic ISBN: 978-1-4704-1084-1
Product Code: SURV/192.E**

List Price: $95.00

AMS Member Price: $76.00

MAA Member Price: $85.50

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#### Supplemental Materials

# Attractors for Degenerate Parabolic Type Equations

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*Messoud Efendiev*

This book deals with the long-time behavior of solutions of degenerate
parabolic dissipative equations arising in the study of biological,
ecological, and physical problems. Examples include porous media
equations, \(p\)-Laplacian and doubly nonlinear equations, as well as
degenerate diffusion equations with chemotaxis and ODE-PDE coupling
systems. For the first time, the long-time dynamics of various
classes of degenerate parabolic equations, both semilinear and
quasilinear, are systematically studied in terms of their global and
exponential attractors.

The long-time behavior of many dissipative systems generated by
evolution equations of mathematical physics can be described in terms
of global attractors. In the case of dissipative PDEs in bounded
domains, this attractor usually has finite Hausdorff and fractal
dimension. Hence, if the global attractor exists, its defining
property guarantees that the dynamical system reduced to the attractor
contains all of the nontrivial dynamics of the original system.
Moreover, the reduced phase space is really “thinner” than
the initial phase space. However, in contrast to nondegenerate
parabolic type equations, for a quite large class of degenerate
parabolic type equations, their global attractors can have infinite
fractal dimension.

The main goal of the present book is to give a detailed and
systematic study of the well-posedness and the dynamics of the
semigroup associated to important degenerate parabolic equations in
terms of their global and exponential attractors. Fundamental topics
include existence of attractors, convergence of the dynamics and the
rate of convergence, as well as the determination of the fractal
dimension and the Kolmogorov entropy of corresponding attractors. The
analysis and results in this book show that there are new effects
related to the attractor of such degenerate equations that cannot be
observed in the case of nondegenerate equations in bounded domains.

This book is published in cooperation with Real Sociedad Matemática Española (RSME)

#### Readership

Graduate students and research mathematicians interested in non-linear PDEs.

#### Reviews & Endorsements

The main aim of this book is to give more insight into such types of PDEs and to fill this gap. This aim is achieved by a systematic study of the well-posedness and the dynamics of the associated semigroup generated by degenerate parabolic equations in terms of their global and exponential attractors as well as studying fractal dimension and Kolmogorov entropy.

-- Zentralblatt Math

#### Table of Contents

# Table of Contents

## Attractors for Degenerate Parabolic Type Equations

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface vii8 free
- Auxiliary materials 112 free
- Global attractors for autonomous evolution equations 1930
- Exponential attractors 2536
- Porous medium equation in homogeneous media: Long-time dynamics 6778
- Porous medium equation in heterogeneous media: Long-time dynamics 89100
- Long-time dynamics of 𝑝-Laplacian equations: Homogeneous-media 101112
- Long-time dynamics of 𝑝-Laplacian equations: Heterogeneous media 107118
- Doubly nonlinear degenerate parabolic equations 125136
- On a class of PDEs with degenerate diffusion and chemotaxis: Autonomous case 147158
- On a class of PDEs with degenerate diffusion and chemotaxis: Nonautonomous case 173184
- ODE-PDE coupling arising in the modelling of a forest ecosystem 191202
- Bibliography 213224
- Index 219230 free
- Back Cover Back Cover1233