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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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