1,006 research outputs found
Radial Fast Diffusion on the Hyperbolic Space
We consider radial solutions to the fast diffusion equation
on the hyperbolic space for , ,
. By radial we mean solutions depending only on the
geodesic distance from a given point . We investigate
their fine asymptotics near the extinction time in terms of a separable
solution of the form , where
is the unique positive energy solution, radial w.r.t. , to for a suitable , a semilinear elliptic problem thoroughly
studied in \cite{MS08}, \cite{BGGV}. We show that converges to in relative error, in the sense that as . In particular the solution is
bounded above and below, near the extinction time , by multiples of
.Comment: To appear in Proc. London Math. So
Fractional porous media equations: existence and uniqueness of weak solutions with measure data
We prove existence and uniqueness of solutions to a class of porous media
equations driven by the fractional Laplacian when the initial data are positive
finite Radon measures on the Euclidean space. For given solutions without a
prescribed initial condition, the problem of existence and uniqueness of the
initial trace is also addressed. By the same methods we can also treat weighted
fractional porous media equations, with a weight that can be singular at the
origin, and must have a sufficiently slow decay at infinity (power-like). In
particular, we show that the Barenblatt-type solutions exist and are unique.
Such a result has a crucial role in [24], where the asymptotic behavior of
solutions is investigated. Our uniqueness result solves a problem left open,
even in the non-weighted case, in [42]Comment: Further results on initial traces added. Some proofs shortene
Uniqueness of very weak solutions for a fractional filtration equation
We prove existence and uniqueness of distributional, bounded, nonnegative
solutions to a fractional filtration equation in . With regards
to uniqueness, it was shown even for more general equations in [19] that if two
bounded solutions of (1.1) satisfy , then . We obtain here that this extra assumption can in
fact be removed and establish uniqueness in the class of merely bounded
solutions, provided they are nonnegative. Indeed, we show that a minimal
solution exists and that any other solution must coincide with it. As a
consequence, distributional solutions have locally-finite energy.Comment: Final version. To appear on Adv. Mat
Conditions at infinity for the inhomogeneous filtration equation
We investigate existence and uniqueness of solutions to the filtration
equation with an inhomogeneous density in , approaching at
infinity a given continuous datum of Dirichlet type.Comment: To appear in Annales de l'Institut Henri Poincar\'e (C) Analyse Non
Lin\'eair
Improved time-decay for a class of scaling critical electromagnetic Schr\"odinger flows
We consider a Schr\"odinger hamiltonian with scaling critical and
time independent external electromagnetic potential, and assume that the
angular operator associated to is positive definite. We prove the
following: if , then
, being a positive number, explicitly depending on the
ground level of and the space dimension . We prove similar results also
for the heat semi-group generated by
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