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Smoothing and Decay Estimates for Nonlinear Diffusion EquationsEquations of Porous Medium Type$
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Juan Luis Vázquez

Print publication date: 2006

Print ISBN-13: 9780199202973

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780199202973.001.0001

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Smoothing effect and time decay from L p or M p

Smoothing effect and time decay from L p or M p

Chapter:
(p.42) 3 Smoothing effect and time decay from L p or M p
Source:
Smoothing and Decay Estimates for Nonlinear Diffusion Equations
Author(s):

Juan Luis Vázquez

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199202973.003.0004

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 results are based on a very delicate phase-plane analysis of the existence of certain types of self-similar solutions. This technique will play a big role in later chapters and the analysis is presented in Section 3.2. The technique allows us to extend the functional setting in a natural way from the Lebesgue spaces into the Marcinkiewicz spaces Mp (R n ). Using this machinery, a special solution is developed in Section 3.3 that replaces the ZKB in the present context and allows us to establish the smoothing effect from Mp into L in Section 3.4. This effect is easily extended into a similar effect that takes place.

Keywords:   smoothing effect, scaling, phase-plane system

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