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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

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

Juan Luis Vázquez

Oxford University Press

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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