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Subject: Mathematics  Book Title: The Porous Medium Equation
The Porous Medium Equation
Mathematical Theory
Vazquez, Juan Luis , Universidad Autónoma de Madrid
Print publication date: 2006
Published to Oxford Scholarship Online: September 2007
Print ISBN-13: 978-0-19-856903-9
doi:10.1093/acprof:oso/9780198569039.001.0001
 
Abstract: The heat equation is one of the three classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it come to be fairly well understood. This book provides a presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation (PME). This equation appears in a number of physical applications, such as to describe processes involving fluid flow, heat transfer, or diffusion. Other applications have been proposed in mathematical biology, lubrication, boundary layer theory, and other fields. Each chapter contains a detailed introduction and is supplied with a section of notes, providing comments, historical notes or recommended reading, and exercises.

Keywords: heat equation, linear partial differential equations, nonlinear, fluid flow, heat transfer, diffusion, mathematical biology, lubrication, boundary layer theory
Table of Contents
Preface
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1. INTRODUCTION
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2. MAIN APPLICATIONS
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3. PRELIMINARIES AND BASIC ESTIMATES
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4. BASIC EXAMPLES
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5. THE DIRICHLET PROBLEM I. WEAK SOLUTIONS
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6. THE DIRICHLET PROBLEM II. LIMIT SOLUTIONS, VERY WEAK SOLUTIONS AND SOME OTHER VARIANTS
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7. CONTINUITY OF LOCAL SOLUTIONS
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8. THE DIRICHLET PROBLEM III. STRONG SOLUTIONS
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9. THE CAUCHY PROBLEM. L -THEORY
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10. THE PME AS AN ABSTRACT EVOLUTION EQUATION. SEMIGROUP APPROACH
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11. THE NEUMANN PROBLEM AND PROBLEMS ON MANIFOLDS
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12. THE CAUCHY PROBLEM WITH GROWING INITIAL DATA
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13. OPTIMAL EXISTENCE THEORY FOR NON-NEGATIVE SOLUTIONS
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14. PROPAGATION PROPERTIES
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15. ONE-DIMENSIONAL THEORY. REGULARITY AND INTERFACES
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16. FULL ANALYSIS OF SELF-SIMILARITY
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17. TECHNIQUES OF SYMMETRIZATION AND CONCENTRATION
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18. ASYMPTOTIC BEHAVIOUR I. THE CAUCHY PROBLEM
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19. REGULARITY AND FINER ASYMPTOTICS IN SEVERAL DIMENSIONS
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20. ASYMPTOTIC BEHAVIOUR II. DIRICHLET AND NEUMANN PROBLEMS
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21. FURTHER APPLICATIONS
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Appendix
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Bibliography
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Index
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doi:10.1093/acprof:oso/9780198569039.001.0001
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PART ONE
PART TWO
COMPLEMENTS