Attractors for Degenerate Parabolic Type Equations

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,  -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).