This chapter addresses the question of extending the theory and estimates of the FDE to cover the singular range m ≤ 0, that has only been partially considered in previous chapters. This completes the scope of the investigation about decay, smoothing effects, and best constants. But the range offers the possibility of learning about a novel and quite interesting dynamical issue: instantaneous extinction, which is tied to boundary layers at the initial time. The second part of the chapter discusses the question of local estimates improving on the results of previous sections. Finally an open problem is presented.
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