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Asymptotics of the porous media equation via Sobolev inequalities.
No full textLet M be a compact Riemannian manifold without boundary. Consider the porous media
equation u_t =\Delta (|u|^(m-1)m), u(0)= u0 ∈ L^q , \Delta being the Laplace–Beltrami operator. Then, for a suitable range of q, the associated evolution is L^q − L^\infty regularizing at any time t >0. For large t it is shown that, for general initial data, u(t) approaches its time-independent mean with quantitative bounds on the rate of convergence. Similar bounds are valid when the manifold is not compact, but u(t) approaches u ≡ 0 with different asymptotics. The case of manifolds with boundary and homogeneous Dirichlet, or Neumann, boundary conditions, is treated as well. The proof stems from a new connection between logarithmic Sobolev inequalities and the contractivity properties
of the nonlinear evolutions considered, and is therefore applicable to a more abstract setting
Direct and reverse Gagliardo-Nirenberg inequalities from logarithmic Sobolev inequalities
No full text
Ultracontractive bounds for nonlinear evolution equations governed by the subcritical -Laplacian
No full textProgr. Nonlin. Differential Equations Vol. 6
Special fast diffusion with slow asymptotics. Entropy method and flow on a Riemannian manifold
No full textWe consider the asymptotic behaviour of positive solutions of the fast diffusion equation u_t = \Delta u^m in the whole Euclidea space R^d, with a precise value for the exponent m = (d − 4)/(d − 2). This case had been left open in the general study (Blanchet et al. in Arch Rat Mech Anal 191:347–385, 2009) 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 suitabl Riemannian Manifold (R^d , g), with a metric g which
is conformal to the standard Rd metric. Studying the pointwise heat kernel behaviour allows to prove suitable Gagliardo–Nirenberg inequalities associated with the generator. Such inequalities in turn allow one 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
Behaviour near extinction for the fast diffusion equation on bounded domains
No full textWe consider the Fast Diffusion Equation posed in a bounded smooth domain Ω ⊂ R^d with homogeneous
Dirichlet conditions. It is known that in the exponent range m_s = (d − 2)_+/(d + 2) < m < 1 all bounded positive solutions
u(t, x) of such problem extinguish in a finite time T = T (u), and also that such solutions approach a separate variable solution
u(t, x) ∼ (T −t)1/(1−m)S(x), as t →T^−.
Here, we are interested in describing the behaviour of the solutions near the extinction time in that range of exponents. We
first show that the convergence v(x, t) = u(t, x)(T − t)^{−1/(1−m)} to S(x) takes place uniformly in the relative error norm. Then,
we study the question of rates of convergence of the rescaled flow, i.e., v →S. For m close to 1 we get such rates by means of
entropy methods and weighted Poincaré inequalities. The analysis of the latter point makes an essential use of fine properties of
a associated stationary elliptic problem in the limit m→1, and such a study has an independent interest
The fractional porous medium equation on the hyperbolic space
Full text linkWe consider a nonlinear degenerate parabolic equation of porous medium type, whose diffusion is driven by the (spectral) fractional Laplacian on the hyperbolic space. We provide existence results for solutions, in an appropriate weak sense, for data belonging either to the usual Lp spaces or to larger (weighted) spaces determined either in terms of a ground state of the laplacian, or of the (fractional) Green’s function. For such solutions, we also prove different kind of smoothing effects, in the form of quantitative L1- L∞ estimates. To the best of our knowledge, this seems the first time in which the fractional porous medium equation has been treated on non-compact, geometrically non-trivial examples
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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