Attractors for degenerate parabolic type equations (Q2855853)

The basic topic studied in the book is the long-time behavior of solutions of degenerate parabolic dissipative equations arising in the study of biological, ecological, and physical problems. The central objects are the global attractors which characterize the long-time behavior of many dissipative systems in the corresponding phase spaces.NEWLINENEWLINE Mainly, the so far existing results for global attractors of evolution partial differential equations (PDEs) have been obtained for PDEs with more or less regular structure (e.g. uniform parabolic). In contrast to this, very little is known concerning the long-time dynamics of degenerate nonlinear parabolic equations, such as porous media equations, \(p\)-Laplacian and doubly nonlinear equations, degenerate diffusion with chemotaxis and ODE-PDE coupling systems and their degenerate extensions.NEWLINENEWLINE 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. It is demonstrated that there are new effects related to the attractors of degenerate PDEs which cannot be observed in the case of nondegenerate equations in bounded domains.NEWLINENEWLINEThe book consists of eleven chapters. The introduction chapter provides us with the asymptotic of the \(\epsilon\)-Kolmogorov entropy in various functional spaces, \(L^p\)-regularity and interior regularity of solutions for degenerate PDEs, embedding theorems and other well-known classical results. Chapter~2 is concerned with the long-time behavior of solutions of PDEs in terms of the attractors, including existence of attractors and their properties. Other important material such as an estimate of time derivatives for nonautonomous perturbations of regular attractors (a cornerstone for developing a new method of proving stabilization to equilibria for solutions of an ODE-PDE coupling problem studied in Chapter~11) and the standard Lojasiewicz technique are exposed. Chapter~3 is devoted to the systematic study of exponential attractors both for autonomous and for nonautonomous dynamical systems. In Chapters~4--7 the author is concerned with the well-posedness (global in time solutions) as well as long-time dynamics (finite and infinite-dimensional) of porous medium and \(p\)-Laplacian equations. In these chapters some new features related to the attractors of such equations, that cannot be observed in nondegenerate cases, are presented. A detailed study of some classes of doubly nonlinear degenerate equations is given in Chapter~8. Also the long-time behavior of solutions of doubly nonlinear degenerate equations is investigated in terms of the associated global and exponential attractors. In Chapters~9 and~10 the author considers both autonomous and nonautonomous chemotaxis systems with degenerate diffusion. The existence and uniqueness results, when the underlying domain is three-dimensional, are given. The main aim of Chapter~11 is to study the long-time behavior of solutions of a class of degenerate parabolic systems describing the development of a forest ecosystem. The problem considered is a coupled system of a second-order ODE with a linear PDE (heat-like equation).NEWLINENEWLINEThe method developed in Chapters~4--11 in order to study the long-time dynamics of certain classes of degenerate parabolic equations can be applied to other classes of degenerate equations, both autonomous and nonautonomous.NEWLINENEWLINEThe book will be interesting for specialists as well as students interested in qualitative analysis of evolution dissipative systems and new methods for degenerate PDEs. The bibliography references are quite complete and contain 97 items.