95 research outputs found
A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on bounded domains
We investigate quantitative properties of the nonnegative solutions
to the nonlinear fractional diffusion equation, , posed in a bounded domain, with for . As we use one of the most common
definitions of the fractional Laplacian , , in a bounded
domain with zero Dirichlet boundary conditions. We consider a general class of
very weak solutions of the equation, and obtain a priori estimates in the form
of smoothing effects, absolute upper bounds, lower bounds, and Harnack
inequalities. We also investigate the boundary behaviour and we obtain sharp
estimates from above and below. The standard Laplacian case or the linear
case are recovered as limits. The method is quite general, suitable to be
applied to a number of similar problems
Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities
In this paper we consider a family of Caffarelli-Kohn-Nirenberg interpolation
inequalities (CKN), with two radial power law weights and exponents in a
subcritical range. We address the question of symmetry breaking: are the
optimal functions radially symmetric, or not ? Our intuition comes from a
weighted fast diffusion (WFD) flow: if symmetry holds, then an explicit entropy
- entropy production inequality which governs the intermediate asymptotics is
indeed equivalent to (CKN), and the self-similar profiles are optimal for
(CKN). We establish an explicit symmetry breaking condition by proving the
linear instability of the radial optimal functions for (CKN). Symmetry breaking
in (CKN) also has consequences on entropy - entropy production inequalities and
on the intermediate asymptotics for (WFD). Even when no symmetry holds in
(CKN), asymptotic rates of convergence of the solutions to (WFD) are determined
by a weighted Hardy-Poincar{\'e} inequality which is interpreted as a
linearized entropy - entropy production inequality. All our results rely on the
study of the bottom of the spectrum of the linearized diffusion operator around
the self-similar profiles, which is equivalent to the linearization of (CKN)
around the radial optimal functions, and on variational methods. Consequences
for the (WFD) flow will be studied in Part II of this work
Weighted fast diffusion equations (Part II): Sharp asymptotic rates of convergence in relative error by entropy methods
This paper is the second part of the study. In Part~I, self-similar solutions
of a weighted fast diffusion equation (WFD) were related to optimal functions
in a family of subcritical Caffarelli-Kohn-Nirenberg inequalities (CKN) applied
to radially symmetric functions. For these inequalities, the linear instability
(symmetry breaking) of the optimal radial solutions relies on the spectral
properties of the linearized evolution operator. Symmetry breaking in (CKN) was
also related to large-time asymptotics of (WFD), at formal level. A first
purpose of Part~II is to give a rigorous justification of this point, that is,
to determine the asymptotic rates of convergence of the solutions to (WFD) in
the symmetry range of (CKN) as well as in the symmetry breaking range, and even
in regimes beyond the supercritical exponent in (CKN). Global rates of
convergence with respect to a free energy (or entropy) functional are also
investigated, as well as uniform convergence to self-similar solutions in the
strong sense of the relative error. Differences with large-time asymptotics of
fast diffusion equations without weights will be emphasized
Quantitative Local Bounds for Subcritical Semilinear Elliptic Equations
The purpose of this paper is to prove local upper and lower bounds for weak
solutions of semilinear elliptic equations of the form , with
, defined on bounded domains of \RR^d, , without
reference to the boundary behaviour. We give an explicit expression for all the
involved constants. As a consequence, we obtain local Harnack inequalities with
explicit constant, as well as gradient bounds.Comment: 2 figure
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