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

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

(p.77) 5 Subcritical range of the FDE. Critical line. Extinction. Backward effect
Smoothing and Decay Estimates for Nonlinear Diffusion Equations

Juan Luis Vázquez

Oxford University Press

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 advancing some of the results, which are valid for the whole subcritical range m < mc , it focuses on the case m > 0. The chapter is organized as follows. Section 5.2 contains the proof of extinction of solutions in the Marcinkiewicz space Mp* (R n), that is characterized as the natural extinction space among all the spaces Mp (R n) and Lp (R n). Section 5.3 considers the question of necessary conditions and the continuity of the extinction time T as a function of u0. Section 5.4 deals with the construction of the global self-similar solutions that increase their space decay rate for positive time. Section 5.5 explains how and where whole mass is lost in the process of extinction. Section 5.6 studies forward effects when dealing with the critical exponent m = mc with starting space L1 . Finally, Section 5.7 discusses extinction as a form of blow-up after a suitable change of variables.

Keywords:   smoothing effect, decay rates, FDE, subcritical range, extinction, critical line

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