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Hardcover ISBN:  9781470409852 
Product Code:  SURV/192 
List Price:  $129.00 
MAA Member Price:  $116.10 
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Product Code:  SURV/192.E 
List Price:  $125.00 
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AMS Member Price:  $100.00 
Hardcover ISBN:  9781470409852 
eBook ISBN:  9781470410841 
Product Code:  SURV/192.B 
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Book DetailsMathematical Surveys and MonographsVolume: 192; 2013; 221 ppMSC: Primary 35; 37
This book deals with the longtime 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 ODEPDE coupling systems. For the first time, the longtime dynamics of various classes of degenerate parabolic equations, both semilinear and quasilinear, are systematically studied in terms of their global and exponential attractors.
The longtime 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 wellposedness 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 Sociedád Matematica Española.ReadershipGraduate students and research mathematicians interested in nonlinear PDEs.

Table of Contents

Chapters

1. Auxiliary materials

2. Global attractors for autonomous evolution equations

3. Exponential attractors

4. Porous medium equation in homogeneous media: Longtime dynamics

5. Porous medium equation in heterogeneous media: Longtime dynamics

6. Longtime dynamics of $p$Laplacian equations: Homogeneousmedia

7. Longtime dynamics of $p$Laplacian equations: Heterogeneous media

8. Doubly nonlinear degenerate parabolic equations

9. On a class of PDEs with degenerate diffusion and chemotaxis: Autonomous case

10. On a class of PDEs with degenerate diffusion and chemotaxis: Nonautonomous case

11. ODEPDE coupling arising in the modelling of a forest ecosystem


Additional Material

Reviews

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 wellposedness 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


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This book deals with the longtime 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 ODEPDE coupling systems. For the first time, the longtime dynamics of various classes of degenerate parabolic equations, both semilinear and quasilinear, are systematically studied in terms of their global and exponential attractors.
The longtime 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 wellposedness 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.
Graduate students and research mathematicians interested in nonlinear PDEs.

Chapters

1. Auxiliary materials

2. Global attractors for autonomous evolution equations

3. Exponential attractors

4. Porous medium equation in homogeneous media: Longtime dynamics

5. Porous medium equation in heterogeneous media: Longtime dynamics

6. Longtime dynamics of $p$Laplacian equations: Homogeneousmedia

7. Longtime dynamics of $p$Laplacian equations: Heterogeneous media

8. Doubly nonlinear degenerate parabolic equations

9. On a class of PDEs with degenerate diffusion and chemotaxis: Autonomous case

10. On a class of PDEs with degenerate diffusion and chemotaxis: Nonautonomous case

11. ODEPDE coupling arising in the modelling of a forest ecosystem

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 wellposedness 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