Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Special fast diffusion with slow asymptotics. Entropy method and flow on
  a Riemannian manifold   We consider the asymptotic behaviour of positive solutions $u(t,x)$ of the
fast diffusion equation $u_t=\Delta (u^{m}/m)={\rm div} (u^{m-1}\nabla u)$
posed for $x\in\RR^d$, $t>0$, with a precise value for the exponent
$m=(d-4)/(d-2)$. The space dimension is $d\ge 3$ so that $m<1$, and even $m=-1$
for $d=3$. This case had been left open in the general study \cite{BBDGV} since
it requires quite different functional analytic methods, due in particular to
the absence of a spectral gap for the operator generating the linearized
evolution.
  The linearization of this flow is interpreted here as the heat flow of the
Laplace-Beltrami operator of a suitable Riemannian Manifold $(\RR^d,{\bf g})$,
with a metric ${\bf g}$ which is conformal to the standard $\RR^d$ metric.
Studying the pointwise heat kernel behaviour allows to prove {suitable
Gagliardo-Nirenberg} inequalities associated to the generator. Such
inequalities in turn allow to study the nonlinear evolution as well, and to
determine its asymptotics, which is identical to the one satisfied by the
linearization. In terms of the rescaled representation, which is a nonlinear
Fokker--Planck equation, the convergence rate turns out to be polynomial in
time. This result is in contrast with the known exponential decay of such
representation for all other values of $m$.

10
The categories of this paper are: q-bio
9
The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg
14
The categories of this paper are: math-ph
25
The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
6
The categories of this paper are: astro-ph
29
The categories of this paper are: funct-an
12
The categories of this paper are: nucl-th
26
The categories of this paper are: bayes-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ex
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The categories of this paper are: physics