Book

Smoothing and Decay Estimates for Nonlinear Diffusion Equations

Juan Luis Vázquez

Published in print August 2006 | ISBN: 9780199202973
Published online September 2007 | e-ISBN: 9780191707919 | DOI: https://dx.doi.org/10.1093/acprof:oso/9780199202973.001.0001

Series: Oxford Lecture Series in Mathematics and Its Applications

Smoothing and Decay Estimates for Nonlinear Diffusion Equations

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Evolution equations of the <i>p</i>-Laplacian type

Juan Luis Vázquez.

in Smoothing and Decay Estimates for Nonlinear Diffusion Equations

August 2006; p ublished online September 2007 .

Chapter. Subjects: Applied Mathematics. 5104 words.

This chapter presents the application of the foregoing ideas to the p-Laplacian evolution equation and the doubly nonlinear evolution equation. Topics covered include the doubly linear...

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Extinction rates and asymptotics for 0 < <i>m</i> < <i>m</i> <sub> <i>c</i> </sub>

Juan Luis Vázquez.

in Smoothing and Decay Estimates for Nonlinear Diffusion Equations

August 2006; p ublished online September 2007 .

Chapter. Subjects: Applied Mathematics. 12969 words.

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...

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Improved analysis of the critical line. Delayed regularity

Juan Luis Vázquez.

in Smoothing and Decay Estimates for Nonlinear Diffusion Equations

August 2006; p ublished online September 2007 .

Chapter. Subjects: Applied Mathematics. 4941 words.

This chapter continues the analysis of the functional properties of the evolution semigroup generated by the FDE on the critical line, i.e., when m < mc and p assumes the...

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Introduction

Juan Luis Vázquez.

in Smoothing and Decay Estimates for Nonlinear Diffusion Equations

August 2006; p ublished online September 2007 .

Chapter. Subjects: Applied Mathematics. 2827 words.

This introductory chapter opens with an explanation of the purpose of the book, which is to obtain basic estimates for some particular nonlinear parabolic equations and to derive...

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Logarithmic diffusion in 2D and intermediate 1D range

Juan Luis Vázquez.

in Smoothing and Decay Estimates for Nonlinear Diffusion Equations

August 2006; p ublished online September 2007 .

Chapter. Subjects: Applied Mathematics. 13962 words.

This chapter studies two transition situations where non-uniqueness plays an important role. The first deals with the range -1 < m ≤ 0 in n = 1, which looks like supercritical but contains...

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Lower bounds, contractivity, error estimates, and continuity

Juan Luis Vázquez.

in Smoothing and Decay Estimates for Nonlinear Diffusion Equations

August 2006; p ublished online September 2007 .

Chapter. Subjects: Applied Mathematics. 7384 words.

This chapter considers a number of bounds that complement the study so far, which was centred on upper bounds. The first section concerns the existence of lower bounds for non-negative...

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Preliminaries

Juan Luis Vázquez.

in Smoothing and Decay Estimates for Nonlinear Diffusion Equations

August 2006; p ublished online September 2007 .

Chapter. Subjects: Applied Mathematics. 6792 words.

This chapter provides necessary preliminaries on functional analysis, comparison results, and the fundamentals of the PME. Section 1.2 provides the basic information on the equation that...

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Smoothing effect and time decay. Data in <i>L</i> <sup>1</sup> (R<sup> <i>n</i> </sup>) or <i>M</i> (R<sup> <i>n</i> </sup>)

Juan Luis Vázquez.

in Smoothing and Decay Estimates for Nonlinear Diffusion Equations

August 2006; p ublished online September 2007 .

Chapter. Subjects: Applied Mathematics. 10171 words.

This chapter discusses the smoothing and decay effects for the porous medium equation, using as a model case the famous Barenblatt solutions that have explicit formulas. Topics covered...

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Smoothing effect and time decay from <i>L</i> <sup> <i>p</i> </sup> or <i>M</i> <sup> <i>p</i> </sup>

Juan Luis Vázquez.

in Smoothing and Decay Estimates for Nonlinear Diffusion Equations

August 2006; p ublished online September 2007 .

Chapter. Subjects: Applied Mathematics. 8908 words.

This chapter addresses the question of boundedness for the same equation when the initial data are chosen in the Lebesgue space Lp, p Ɛ (1, ∞). It is assumed that m > mc . The...

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Subcritical range of the FDE. Critical line. Extinction. Backward effect

Juan Luis Vázquez.

in Smoothing and Decay Estimates for Nonlinear Diffusion Equations

August 2006; p ublished online September 2007 .

Chapter. Subjects: Applied Mathematics. 16095 words.

This chapter studies the smoothing effect and decay rates for the FDE in the subcritical range m < mc , and also for the critical exponent mc = (n - 2)/n. While...

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