1,061 research outputs found

### Radial Fast Diffusion on the Hyperbolic Space

We consider radial solutions to the fast diffusion equation $u_t=\Delta u^m$
on the hyperbolic space $\mathbb{H}^{N}$ for $N \ge 2$, $m\in(m_s,1)$,
$m_s=\frac{N-2}{N+2}$. By radial we mean solutions depending only on the
geodesic distance $r$ from a given point $o \in \mathbb{H}^N$. We investigate
their fine asymptotics near the extinction time $T$ in terms of a separable
solution of the form ${\mathcal V}(r,t)=(1-t/T)^{1/(1-m)}V^{1/m}(r)$, where $V$
is the unique positive energy solution, radial w.r.t. $o$, to $-\Delta
V=c\,V^{1/m}$ for a suitable $c>0$, a semilinear elliptic problem thoroughly
studied in \cite{MS08}, \cite{BGGV}. We show that $u$ converges to ${\mathcal
V}$ in relative error, in the sense that $\|{u^m(\cdot,t)}/{{\mathcal
V}^m(\cdot,t)}-1\|_\infty\to0$ as $t\to T^-$. In particular the solution is
bounded above and below, near the extinction time $T$, by multiples of
$(1-t/T)^{1/(1-m)}e^{-(N-1)r/m}$.Comment: To appear in Proc. London Math. So

### Weighted dispersive estimates for two-dimensional Schr\"odinger operators with Aharonov-Bohm magnetic field

We consider two-dimensional Schr\"odinger operators $H$ with Aharonov-Bohm
magnetic field and an additional electric potential. We obtain an explicit
leading term of the asymptotic expansion of the unitary group $e^{-i t H}$ for
$t\to\infty$ in weighted $L^2$ spaces. In particular, we show that the magnetic
field improves the decay of $e^{-i t H}$ with respect to the unitary group
generated by non-magnetic Schr\"odinger operators, and that the decay rate in
time is determined by the magnetic flux.Comment: To appear in J. Differential Equation

### 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 ${\mathbb R}^d$. With regards
to uniqueness, it was shown even for more general equations in [19] that if two
bounded solutions $u,w$ of (1.1) satisfy $u-w\in L^1({\mathbb
R}^d\times(0,T))$, then $u=w$. 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 ${\mathbb R}^N$, 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

### Sharp Poincar\'e-Hardy and Poincar\'e-Rellich inequalities on the hyperbolic space

We study Hardy-type inequalities associated to the quadratic form of the
shifted Laplacian $-\Delta_{\mathbb H^N}-(N-1)^2/4$ on the hyperbolic space
${\mathbb H}^N$, $(N-1)^2/4$ being, as it is well-known, the bottom of the
$L^2$-spectrum of $-\Delta_{\mathbb H^N}$. We find the optimal constant in the
resulting Poincar\'e-Hardy inequality, which includes a further remainder term
which makes it sharp also locally. A related inequality under suitable
curvature assumption on more general manifolds is also shown. Similarly, we
prove Rellich-type inequalities associated with the shifted Laplacian, in which
at least one of the constant involved is again sharp.Comment: Final version. To appear in JF

### Sharp two-sided heat kernel estimates of twisted tubes and applications

We prove on-diagonal bounds for the heat kernel of the Dirichlet Laplacian
$-\Delta^D_\Omega$ in locally twisted three-dimensional tubes $\Omega$. In
particular, we show that for any fixed $x$ the heat kernel decays for large
times as $\mathrm{e}^{-E_1t}\, t^{-3/2}$, where $E_1$ is the fundamental
eigenvalue of the Dirichlet Laplacian on the cross section of the tube. This
shows that any, suitably regular, local twisting speeds up the decay of the
heat kernel with respect to the case of straight (untwisted) tubes. Moreover,
the above large time decay is valid for a wide class of subcritical operators
defined on a straight tube. We also discuss some applications of this result,
such as Sobolev inequalities and spectral estimates for Schr\"odinger operators
$-\Delta^D_\Omega-V$.Comment: To appear in Arch. Rat. Mech. Ana

- â€¦