1:00 pm Wednesday, March 10, 2010
Analysis Seminar : Positivity, Local Smoothing Effects and Harnack Inequalities for Very Fast Diffusion Equations by
Matteo Bonforte (Universidad Autonoma de Madrid) in RLM 10.176
We investigate qualitative properties of local solutions to the fast diffusion equation, with , and with , corresponding to general nonnegative initial data. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of . They combine into forms of new Harnack inequalities that are typical of fast diffusion equations. Such results are new for and in the so-called very fast diffusion range, precisely for all , and/or for all , where is the dimension of the Euclidean space . In the supercritical case we recover the (sharp) results existing in literature with a different proof. The boundedness results are true even for , or , while the positivity ones cannot be true in that range. For the fast -Laplacian we also prove a new local energy inequality for suitable norms of the gradients of the solutions, which can be extended to more general operators of -Laplacian type. As a consequence, we show that bounded local weak solutions are indeed local strong solutions for any , more precisely . Submitted by
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