Rate of Convergence to Barenblatt Profiles for the Fast Diffusion Equation

We study the asymptotic behaviour of positive solutions of the Cauchy problem for the fast diffusion equation near the extinction time. We find a continuum of rates of convergence to a self-similar profile. These rates depend explicitly on the spatial decay rates of initial data

The fractional porous medium equation on the hyperbolic space

We 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

Local smoothing effects, positivity, and Harnack inequalities for the fast p-Laplacian equation

We study qualitative and quantitative properties of local weak solutions of the fast $p$-Laplacian equation, $\partial_t u=\Delta_{p}u$, with $1<p<2$. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of \RR^n\times [0,T]. We combine these lower and upper bounds in different forms of intrinsic Harnack inequalities, which are new in the very fast diffusion range, that is when $1<p \le 2n/(n+1)$. The boundedness results may be also extended to the limit case $p=1$, while the positivity estimates cannot. We prove the existence as well as sharp asymptotic estimates for the so-called large solutions for any $1<p<2$, and point out their main properties. We also prove a new local energy inequality for suitable norms of the gradients of the solutions. As a consequence, we prove that bounded local weak solutions are indeed local strong solutions, more precisely $\partial_t u\in L^2_{\rm loc}$

Ground deformations related to the effusive eruptions of Stromboli: the 2002-2003 case

Stromboli volcano erupted suddenly on 28 December 2002 after a long period of typically persistent and moderate explosive activity. Lava flows outpoured from the northern wall of the NE crater and descended into the Sciara del Fuoco (SdF). On December 30th, 2002, two landslides occurred on the northern part of the SdF, producing a tsunami that caused significant damage. This event led to the upgrading of the ground deformation monitoring system. The new requisite was the real-time detection of the deformation related both to the magma movements within the eruptive feeding system and to potential slope failures of the SdF. To this end, a remotely controlled monitoring system, based both on high-frequency (1 Hz) instantaneous GPS and terrestrial geodetic techniques (manual EDM measurements, transformed in automated terrestrial geodetic measurements) was planned and set up in a few months. During the recorded eruptive phases, the new monitoring system aided the Department of Civil Protection in making decisions related to hazards from landslides and volcanic activity and, more generally, on the evolution of volcanic phenomena throughout the eruption. The measurements carried out on the benchmarks located on the high flank allowed us to make some hypotheses on the dynamics of the craters. In particular, the behaviour of the EDM baselines, showing alternating periods of increase and periods of stop in length variation, could be linked to movements of the magmatic column within the craters. Moreover, the monitoring system gave us the opportunity to observe the effects of an effusive vent opening on February 16th. The new geodetic network provided, for the first time, useful information on ground deformations due to shallow and very shallow volcanic sources at Stromboli

Atmospheric anomalies over Mt.Etna using GPS signal delays and tomography of radio wave velocities

Due to the prominent topography of Mt. Etna, the use of satellite geodetic techniques may significantly suffer from atmospheric heterogeneities. This problem mainly affects the DInSAR technique. To overcome these drawbacks the present study attempts to make headway in measuring and interpreting atmospheric anomalies. We used the GAMIT software to obtain the ZTD (Zenith Total Delay) values for the GPS sessions performed on 1996-97, during ERS-2 passes at Mt. Etna. GAMIT software also allows to characterize the statistical behaviour of the tropospheric effects, by using residuals for each station-satellite pair, and to locate the atmospheric anomalies, present mostly at low altitudes. The attempt at using these results to produce a tomography of radio waves velocity of the troposphere suggests that the number of GPS stations used to investigate atmosphere is a critical point in such a study. The three stations are too few to invert anomalies eventually existing in the lower atmosphere. This result is a good starting point for better direct future study to verify the applicability of this tomographic technique to a geodetic network with a higher number of stations, with the aim of characterizing the lower atmosphere of Mt. Etna for a more accurate monitoring of ground deformations

Ground deformation modeling of flank dynamics prior to the 2002 eruption of Mt. Etna

On 22 September 2002, 1 month before the beginning of the flank eruption on the NE Rift, an M-3.7 earthquake struck the northeastern part of Mt. Etna, on the westernmost part of the Pernicana fault. In order to investigate the ground deformation pattern associated with this event, a multi-disciplinary approach is presented here. Just after the earthquake, specific GPS surveys were carried out on two small sub-networks, aimed at monitoring the eastern part of the Pernicana fault, and some baselines belonging to the northeastern EDM monitoring network of Mt. Etna were measured. The leveling route on the northeastern flank of the volcano was also surveyed. Furthermore, an investigation using SAR interferometry was performed and also the continuous tilt data recorded at a high precision sensor close to the epicenter were analyzed to constrain the coseismic deformation. The results of the geodetic surveys show a ground deformation pattern that affects the entire northeastern flank of the volcano, clearly shaped by the Pernicana fault, but too strong and wide to be related only to an M-3.7 earthquake. Leveling and DInSAR data highlight a local strong subsidence, up to 7 cm, close to the Pernicana fault. Significant displacements, up to 2 cm, were also detected on the upper part of the NE Rift and in the summit craters area, while the displacements decrease at lower altitude, suggesting that the dislocation did not continue further eastward. Three-dimensional GPS data inversions have been attempted in order to model the ground deformation source and its relationship with the volcano plumbing system. The model has also been constrained by vertical displacements measured by the leveling survey and by the deformation map obtained by SAR interferometry

Remote Sensing and Geodetic Measurements for Volcanic Slope Monitoring: Surface Variations Measured at Northern Flank of La Fossa Cone (Vulcano Island, Italy)

Abstract: Results of recent monitoring activities on potentially unstable areas of the NW volcano flank of La Fossa cone (Vulcano Island, Italy) are shown here. They are obtained by integration of data by aerial photogrammetry, terrestrial laser scanning (TLS) and GPS taken in the 1996–2011 time span. A comparison between multi-temporal models built from remote sensing data (photogrammetry and TLS) highlights areas characterized by ~7–10 cm/y positive differences (i.e., elevation increase) in the upper crown of the slope. The GPS measurements confirm these results. Areas characterized by negative differences, related to both mass collapses or small surface lowering, also exist. The higher differences, positive and negative, are always observed in zones affected by higher fumarolic activity. In the 2010–2012 time span, ground motions in the northern part of the crater rim, immediately above the upper part of observed area, are also observed. The results show different trends for both vertical and horizontal displacements of points distributed along the rim, with a magnitude of some centimeters, thus revealing a complex kinematics. A slope stability analysis shows that the safety factors estimated from these data do not OPEN ACCESS Remote Sens. 2013, 5 2239 indicate evidence of possible imminent failures. Nevertheless, new time series are needed to detect possible changes with the time of the stability conditions, and the monitoring has to go on

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.Comment: To appear in DCDS-

Special fast diffusion with slow asymptotics. Entropy method and flow on a Riemannian manifold

We consider the asymptotic behaviour of positive solutions $u(t,x)$ of the fast diffusion equation $u_t=\Delta (u^{m}/m)={\rm div} (u^{m-1}\nabla u)$ posed for x\in\RR^d, $t>0$, with a precise value for the exponent $m=(d-4)/(d-2)$. The space dimension is $d\ge 3$ so that $m<1$, and even $m=-1$ for $d=3$. This case had been left open in the general study \cite{BBDGV} 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 suitable Riemannian Manifold (\RR^d,{\bf g}), with a metric ${\bf g}$ which is conformal to the standard \RR^d metric. Studying the pointwise heat kernel behaviour allows to prove {suitable Gagliardo-Nirenberg} inequalities associated to the generator. Such inequalities in turn allow 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 $m$.Comment: 37 page

Kinematics of the Central Mediterranean Plate Boundary, Internal Deformation of Sicily and Interseismic Strain Accumulation Across the Messina Straits

In this work we present a new velocity field, obtained by analyzing continuous GPS (CGPS) stations operating in the Mediterranean area (updated to September 2007)and epoch GPS (EGPS) stations in the 1991-2006 time span, particularly denser in the Iblean plateau and across the Messina Straits