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Gabriele Grillo

Publications and source records attributed to Gabriele Grillo.

At least 37 records · Page 2Linked to original sources
arXiv2020

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.

math.AP↗
arXiv2018

Improved Multipolar Poincaré-Hardy inequalities on Cartan-Hadamard Manifolds

We prove a family of improved multipolar Poincaré-Hardy inequalities on Cartan-Hadamard manifolds. For suitable configurations of poles, these inequalities yield an improved multipolar Hardy inequality and an improved multipolar Poincaré inequality such that the critical unipolar singular mass is reached at any pole.

math.AP↗
arXiv2018

The porous medium equation on Riemannian manifolds with negative curvature: the superquadratic case

We study the long-time behaviour of nonnegative solutions of the Porous Medium Equation posed on Cartan-Hadamard manifolds having very large negative curvature, more precisely when the sectional or Ricci curvatures diverge at infinity more than quadratically in terms of the geodesic distance to the pole. We find an unexpected separate-variable behaviour that reminds one of Dirichlet problems on bounded Euclidean domains. As a crucial step, we prove existence of solutions to a related sublinear elliptic problem, a result of independent interest. Uniqueness of solutions vanishing at infinity is also shown, along with comparison principles, both in the parabolic and in the elliptic case. Our results complete previous analyses of the Porous Medium Equation flow on negatively curved Riemannian manifolds, which were carried out first for the hyperbolic space and then for general Cartan-Hadamard manifolds with a negative curvature having at most quadratic growth. We point out that no similar analysis seems to exist for the linear heat flow. We also translate such results into some weighted Porous Medium Equations in the Euclidean space having special weights.

math.AP↗
arXiv2018

Blow-up and global existence for the porous medium equation with reaction on a class of Cartan-Hadamard manifolds

We consider the porous medium equation with power-type reaction terms $u^p$ on negatively curved Riemannian manifolds, and solutions corresponding to bounded, nonnegative and compactly supported data. If $p>m$, small data give rise to global-in-time solutions while solutions associated to large data blow up in finite time. If $p<m$, large data blow up at worst in infinite time, and under the stronger restriction $p\in(1,(1+m)/2]$ all data give rise to solutions existing globally in time, whereas solutions corresponding to large data blow up in infinite time. The results are in several aspects significantly different from the Euclidean ones, as has to be expected since negative curvature is known to give rise to faster diffusion properties of the porous medium equation.

math.AP↗
arXiv2016

Sharp Poincaré-Hardy and Poincaré-Rellich inequalities on the hyperbolic space

We study Hardy-type inequalities associated to the quadratic form of the shifted Laplacian $-Δ_{\mathbb H^N}-(N-1)^2/4$ 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 $-Δ_{\mathbb H^N}$. We find the optimal constant in the resulting Poincaré-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.

math.CA↗
arXiv2016

The porous medium equation with measure data on negatively curved Riemannian manifolds

We investigate existence and uniqueness of weak solutions of the Cauchy problem for the porous medium equation on negatively curved Riemannian manifolds. We show existence of solutions taking as initial condition a finite Radon measure, not necessarily positive. We then establish uniqueness in the class of nonnegative solutions. On the other hand, we prove that any weak solution of the porous medium equation necessarily takes on as initial datum a finite Radon measure. In addition, we obtain some results in potential analysis on manifolds, concerning the validity of a modified version of the mean-value inequality for superharmonic functions, and properties of potentials of positive Radon measures. Such results are new and of independent interest, and are crucial for our approach.

math.AP↗
arXiv2016

The porous medium equation with large initial data on negatively curved Riemannian manifolds

We show existence and uniqueness of very weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds satisfying suitable lower bounds on Ricci curvature, with initial data that can grow at infinity at a prescribed rate, that depends crucially on the curvature bounds. The curvature conditions we require are sharp for uniqueness in the sense that if they are not satisfied then, in general, there can be infinitely many solutions of the Cauchy problem even for bounded data. Furthermore, under matching upper bounds on sectional curvatures, we give a precise estimate for the maximal existence time, and we show that in general solutions do not exist if the initial data grow at infinity too fast. This proves in particular that the growth rate of the data we consider is optimal for existence. Pointwise blow-up is also shown for a particular class of manifolds and of initial data.

math.AP↗
arXiv2016

The porous medium equation on Riemannian manifolds with negative curvature. The large-time behaviour

We consider nonnegative solutions of the porous medium equation (PME) on a Cartan-Hadamard manifold whose negative curvature can be unbounded. We take compactly supported initial data because we are also interested in free boundaries. We classify the geometrical cases we study into quasi-hyperbolic, quasi-Euclidean and critical cases, depending on the growth rate of the curvature at infinity. We prove sharp upper and lower bounds on the long-time behaviour of the solutions in terms of corresponding bounds on the curvature. In particular we obtain a sharp form of the smoothing effect on such manifolds. We also estimate the location of the free boundary. A global Harnack principle follows. We also present a change of variables that allows to transform radially symmetric solutions of the PME on model manifolds into radially symmetric solutions of a corresponding weighted PME on Euclidean space and back. This equivalence turns out to be an important tool of the theory.

math.AP↗
arXiv2015

Smoothing effects for the porous medium equation on Cartan-Hadamard manifolds

We prove three sharp bounds for solutions to the porous medium equation posed on Riemannian manifolds, or for weighted versions of such equation. Firstly we prove a smoothing effect for solutions which is valid on any Cartan-Hadamard manifold whose sectional curvatures are bounded above by a strictly negative constant. This bound includes as a special case the sharp smoothing effect recently proved by Vázquez on the hyperbolic space in \cite{V}, which is similar to the absolute bound valid in the case of bounded Euclidean domains but has a logarithmic correction. Secondly we prove a bound which interpolates between such smoothing effect and the Euclidean one, supposing a suitable quantitative Sobolev inequality holds, showing that it is sharp by means of explicit examples. Finally, assuming a stronger functional inequality of sub-Poincaré type, we prove that the above mentioned (sharp) absolute bound holds, and provide examples of weighted porous media equations on manifolds of infinite volume in which it holds, in contrast with the non-weighted Euclidean situation. It is also shown that sub-Poincaré inequalities cannot hold on Cartan-Hadamard manifolds.

math.AP↗
arXiv2015

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]

math.AP↗
arXiv2015

Improved time-decay for a class of scaling critical electromagnetic Schrödinger flows

We consider a Schrödinger hamiltonian $H(A,a)$ with scaling critical and time independent external electromagnetic potential, and assume that the angular operator $L$ associated to $H$ is positive definite. We prove the following: if $\|e^{-itH(A,a)}\|_{L^1\to L^\infty}\lesssim t^{-n/2}$, then $ \||x|^{-g(n)}e^{-itH(A,a)}|x|^{-g(n)}\|_{L^1\to L^\infty}\lesssim t^{-n/2-g(n)}$, $g(n)$ being a positive number, explicitly depending on the ground level of $L$ and the space dimension $n$. We prove similar results also for the heat semi-group generated by $H(A,a)$.

math.AP↗
arXiv2014

On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density

We are concerned with the long time behaviour of solutions to the fractional porous medium equation with a variable spatial density. We prove that if the density decays slowly at infinity, then the solution approaches the Barenblatt-type solution of a proper singular fractional problem. If, on the contrary, the density decays rapidly at infinity, we show that the minimal solution multiplied by a suitable power of the time variable converges to the minimal solution of a certain fractional sublinear elliptic equation.

math.AP↗
arXiv2014

Weighted dispersive estimates for two-dimensional Schrödinger operators with Aharonov-Bohm magnetic field

We consider two-dimensional Schrödinger 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 Schrödinger operators, and that the decay rate in time is determined by the magnetic flux.

math.SP↗
arXiv2014

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

We prove on-diagonal bounds for the heat kernel of the Dirichlet Laplacian $-Δ^D_Ω$ in locally twisted three-dimensional tubes $Ω$. 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 Schrödinger operators $-Δ^D_Ω-V$.

math.AP↗
arXiv2013

Radial Fast Diffusion on the Hyperbolic Space

We consider radial solutions to the fast diffusion equation $u_t=Δu^m$ 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 $-ΔV=c\,V^{1/m}$ 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 $(1-t/T)^{1/(1-m)}e^{-(N-1)r/m}$.

math.AP↗
arXiv2012

Porous media equations with two weights: smoothing and decay properties of energy solutions via Poincaré inequalities

We study weighted porous media equations on domains $Ω\subseteq{\mathbb R}^N$, either with Dirichlet or with Neumann homogeneous boundary conditions when $Ω\not={\mathbb R}^N$. Existence of weak solutions and uniqueness in a suitable class is studied in detail. Moreover, $L^{q_0}$-$L^\varrho$ smoothing effects ($1\leq q_0<\varrho<\infty$) are discussed for short time, in connection with the validity of a Poincaré inequality in appropriate weighted Sobolev spaces, and the long-time asymptotic behaviour is also studied. Particular emphasis is given to the Neumann problem, which is much less studied in the literature, as well as to the case $Ω={\mathbb R}^N$ when the corresponding weight makes its measure finite, so that solutions converge to their weighted average instead than to zero. Examples are given in terms of wide classes of weights.

math.AP↗
arXiv2012

Stability and qualitative properties of radial solutions of the Lane-Emden-Fowler equation on Riemannian models

We study existence, uniqueness and stability of radial solutions of the Lane-Emden-Fowler equation $-Δ_g u=|u|^{p-1}u$ in a class of Riemannian models $(M,g)$ of dimension $n\ge 3$ which includes the classical hyperbolic space $\mathbb H^n$ as well as manifolds with sectional curvatures unbounded below. Sign properties and asymptotic behavior of solutions are influenced by the critical Sobolev exponent while the so-called Joseph-Lundgren exponent is involved in the stability of solutions.

math.DG↗