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

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

This chapter studies the smoothing effect and decay rates for the FDE in the subcritical range *m* < *m _{c} *, and also for the critical exponent

*m*= (n - 2)/n. While advancing some of the results, which are valid for the whole subcritical range

_{c}*m*<

*m*, it focuses on the case

_{c}*m*> 0. The chapter is organized as follows. Section 5.2 contains the proof of extinction of solutions in the Marcinkiewicz space

*M*(

^{p*}**R**

^{n}), that is characterized as the natural extinction space among all the spaces

*M*(

^{p}**R**

^{n}) and

*L*(

^{p}**R**

^{n}). Section 5.3 considers the question of necessary conditions and the continuity of the extinction time

*T*as a function of u

_{0}. 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*=

*m*with starting space

_{c}*L*. Finally, Section 5.7 discusses extinction as a form of blow-up after a suitable change of variables.

^{1}*Keywords:*
smoothing effect, decay rates, FDE, subcritical range, extinction, critical line

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