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## THE DIRICHLET PROBLEM I. WEAK 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/9780198569039.003.0005
- Subject:
- Mathematics, Mathematical Physics

This chapter begins with a systematic study of the questions of existence, uniqueness, and main properties of the solutions of the PME by concentrating on the first boundary-value problem posed in a ... 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

## THE DIRICHLET PROBLEM II. LIMIT SOLUTIONS, VERY WEAK SOLUTIONS AND SOME OTHER VARIANTS

*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/9780198569039.003.0006
- Subject:
- Mathematics, Mathematical Physics

This chapter continues the analysis of the initial and boundary value problem. Topics covered include limit solutions, theory of very weak solutions, problems in different domains, limit solutions to ... More

## THE DIRICHLET PROBLEM III. STRONG 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/9780198569039.003.0008
- Subject:
- Mathematics, Mathematical Physics

This chapter addresses the question of how regular the solutions constructed in previous chapters actually are. Section 8.1 considers the question of further regularity of the time derivative ut ... More

## CONTINUITY OF LOCAL 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/9780198569039.003.0007
- Subject:
- Mathematics, Mathematical Physics

This chapter addresses a main issue of the theory, namely, the continuity of the solutions for times t > 0. The equation treated in this chapter is a generalized version of the GPME. The question of ... More

## Extinction rates and asymptotics for 0 < m < m c

*Juan Luis Vázquez*

### in Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780199202973
- eISBN:
- 9780191707919
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199202973.003.0008
- Subject:
- Mathematics, Applied Mathematics

This chapter deals with the actual asymptotic behaviour of the solutions of the FDE in the exponent range 0 < m < mc . This behaviour depends on the class of initial data. The ... More

## Probability Theory and Partial Differential Equations

*Gopinath Kallianpur and P. Sundar*

### in Stochastic Analysis and Diffusion Processes

- Published in print:
- 2014
- Published Online:
- April 2014
- ISBN:
- 9780199657063
- eISBN:
- 9780191781759
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199657063.003.0008
- Subject:
- Mathematics, Probability / Statistics, Applied Mathematics

The connection between stochastic differential equations and partial differential equations is discussed in this chapter. The Dirichlet problem is studied and several examples are presented. Next, ... More

## Generalized Dirichlet and Neumann Boundary-Value Problems

*D. E. Edmunds and W. D. Evans*

### in Spectral Theory and Differential Operators

- Published in print:
- 2018
- Published Online:
- September 2018
- ISBN:
- 9780198812050
- eISBN:
- 9780191861130
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198812050.003.0006
- Subject:
- Mathematics, Pure Mathematics

In this chapter, the generalized or weak interpretation of the Dirichlet and Neumann problems for general elliptic expressions is motivated and then the Lax–Milgram Theorem is used to set the ... More

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