## THE PME AS AN ABSTRACT EVOLUTION EQUATION. SEMIGROUP APPROACH

*Juan Luis Vázquez*

### in The Porous Medium Equation: Mathematical Theory

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198569039
- eISBN:
- 9780191717468
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198569039.003.0010
- Subject:
- Mathematics, Mathematical Physics

This chapter addresses the problem of the construction of solutions of the GPME by viewing it as an abstract evolution equation, or more precisely as an ordinary differential equation with values in ... More

## THE NEUMANN PROBLEM AND PROBLEMS ON MANIFOLDS

*Juan Luis Vázquez*

### in The Porous Medium Equation: Mathematical Theory

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198569039
- eISBN:
- 9780191717468
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198569039.003.0011
- Subject:
- Mathematics, Mathematical Physics

This chapter completes the investigation of previous chapters on the Dirichlet and Cauchy problems by applying the techniques to other important problems. It selects two directions, the Neumann ... More

## THE CAUCHY PROBLEM WITH GROWING INITIAL DATA

*Juan Luis Vázquez*

### in The Porous Medium Equation: Mathematical Theory

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198569039
- eISBN:
- 9780191717468
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198569039.003.0012
- Subject:
- Mathematics, Mathematical Physics

This chapter examines general conditions on the initial data under which the PME can be solved. The emphasis is on non-negative solutions and the data are locally integrable functions. Signed ... More

## OPTIMAL EXISTENCE THEORY FOR NON-NEGATIVE SOLUTIONS

*Juan Luis Vázquez*

### in The Porous Medium Equation: Mathematical Theory

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198569039
- eISBN:
- 9780191717468
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198569039.003.0013
- Subject:
- Mathematics, Mathematical Physics

This chapter studies the existence and uniqueness of solutions of the Cauchy problem for the PME posed in the whole space, which take a Radon measure as initial data. Section 13.1 constructs limit ... More

## ONE-DIMENSIONAL THEORY. REGULARITY AND INTERFACES

*Juan Luis Vázquez*

### in The Porous Medium Equation: Mathematical Theory

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198569039
- eISBN:
- 9780191717468
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198569039.003.0015
- Subject:
- Mathematics, Mathematical Physics

This chapter presents an introduction to some of the main topics relating to the PME, focusing on non-negative solutions. Section 15.1 presents a detailed analysis of the regularity of the pressure, ... More

## FULL ANALYSIS OF SELF-SIMILARITY

*Juan Luis Vázquez*

### in The Porous Medium Equation: Mathematical Theory

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198569039
- eISBN:
- 9780191717468
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198569039.003.0016
- Subject:
- Mathematics, Mathematical Physics

This chapter examines the class of solutions of the PME that are invariant under the scaling group in the variables (x, t, u), and take therefore the so-called self-similar form. After a detailed ... More

## TECHNIQUES OF SYMMETRIZATION AND CONCENTRATION

*Juan Luis Vázquez*

### in The Porous Medium Equation: Mathematical Theory

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198569039
- eISBN:
- 9780191717468
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198569039.003.0017
- Subject:
- Mathematics, Mathematical Physics

This chapter aims to obtain basic estimates for the PME and related equations using as a main tool the techniques of symmetrization and mass concentration comparison, combined with scaling ... More

## ASYMPTOTIC BEHAVIOUR I. THE CAUCHY PROBLEM

*Juan Luis Vázquez*

### in The Porous Medium Equation: Mathematical Theory

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198569039
- eISBN:
- 9780191717468
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198569039.003.0018
- Subject:
- Mathematics, Mathematical Physics

This chapter begins with a study of the behaviour of solutions of the PME for large times. The cornerstone of the presentation is the interplay between asymptotic behaviour and self-similarity. It is ... More

## REGULARITY AND FINER ASYMPTOTICS IN SEVERAL DIMENSIONS

*Juan Luis Vázquez*

### in The Porous Medium Equation: Mathematical Theory

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198569039
- eISBN:
- 9780191717468
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198569039.003.0019
- Subject:
- Mathematics, Mathematical Physics

This chapter investigates the precise regularity that is to be expected for non-negative solutions of the PME that have free boundaries. Section 19.1 describes the results of Caffarelli, Vázquez, and ... More

## ASYMPTOTIC BEHAVIOUR II. DIRICHLET AND NEUMANN PROBLEMS

*Juan Luis Vázquez*

### in The Porous Medium Equation: Mathematical Theory

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198569039
- eISBN:
- 9780191717468
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198569039.003.0020
- Subject:
- Mathematics, Mathematical Physics

This chapter contains a complete study of the large-time behaviour of solutions of the porous medium equation, ut = Δum with m > 1, posed in a bounded ... More