Spectral gap global solutions for degenerate Kirchhoff equations

We consider the second order Cauchy problem $$u''+m(|A^{1/2}u|^2)Au=0, u(0)=u_{0}, u'(0)=u_{1},$$ where $m:[0,+\infty)\to[0,+\infty)$ is a continuous function, and $A$ is a self-adjoint nonnegative operator with dense domain on a Hilbert space. It is well known that this problem admits local-in-time solutions provided that $u_{0}$ and $u_{1}$ are regular enough, depending on the continuity modulus of $m$, and on the strict/weak hyperbolicity of the equation. We prove that for such initial data $(u_{0},u_{1})$ there exist two pairs of initial data $(\overline{u}_{0},\overline{u}_{1})$, $(\widehat{u}_{0},\widehat{u}_{1})$ for which the solution is global, and such that $u_{0}=\overline{u}_{0}+\widehat{u}_{0}$, $u_{1}=\overline{u}_{1}+\widehat{u}_{1}$. This is a byproduct of a global existence result for initial data with a suitable spectral gap, which extends previous results obtained in the strictly hyperbolic case with a smooth nonlinearity $m$.Comment: 16 page

Global Strichartz estimates for an inhomogeneous Maxwell system

We show a global in time Strichartz estimate for the isotropic Maxwell system with divergence free data. On the scalar permittivity and permeability we impose decay assumptions as $|x|\rightarrow\infty$ and a non-trapping condition. The proof is based on smoothing estmates in weighted $L^2$ spaces which follow from corresponding resolvent estimates for the underlying Helmholtz problem

Global well-posedness of the Kirchhoff equation and Kirchhoff systems

This article is devoted to review the known results on global well-posedness for the Cauchy problem to the Kirchhoff equation and Kirchhoff systems with small data. Similar results will be obtained for the initial-boundary value problems in exterior domains with compact boundary. Also, the known results on large data problems will be reviewed together with open problems.Comment: arXiv admin note: text overlap with arXiv:1211.300

Evidence-based urology in practice : heterogeneity in a systematic review meta-analysis

The definitive version is available at www3.interscience.wiley.com.Peer reviewedPreprin

La Nostra Vita con Ezio e Ricordi di guerra

Our life with Ezio and Memories of War, written by Flora Aghib Levi D’Ancona traces the life of her husband Ezio Levi, a Jewish Italian philologist and hispanist, their experiences of exile in the US where the couple fled after the racial laws. Completed with a historiographical introduction and an appendix of unpublished letters, the volume traces Ezio’s path as a Jewish intellectual in Fascist Italy, his role as a cultural mediator of Spanish contemporary literature to Italy, the trauma of the racial laws, and the challenges of the American exile. Expression of a women’s exile literature, the pages reflect the authors experience as a mother writing for her children left in Italy and of an intellectual Italian Jewish woman dealing with the challenges of exile and memory

Dispersive estimates for Schr\"odinger operators with point interactions in R3\mathbb{R}^3

The study of dispersive properties of Schr\"odinger operators with point interactions is a fundamental tool for understanding the behavior of many body quantum systems interacting with very short range potential, whose dynamics can be approximated by non linear Schr\"odinger equations with singular interactions. In this work we proved that, in the case of one point interaction in $\mathbb{R}^3$, the perturbed Laplacian satisfies the same $L^p-L^q$ estimates of the free Laplacian in the smaller regime $q\in[2,3)$. These estimates are implied by a recent result concerning the $L^p$ boundedness of the wave operators for the perturbed Laplacian. Our approach, however, is more direct and relatively simple, and could potentially be useful to prove optimal weighted estimates also in the regime $q\geq 3$.Comment: To appear on: "Advances in Quantum Mechanics: Contemporary Trends and Open Problems", G. Dell'Antonio and A. Michelangeli eds., Springer-INdAM series 201

On the flows associated to selfadjoint operators on metric measure spaces

Let X be a metric space with a doubling measure satisfying μ(B)≳rBn for any ball B with any radius rB> 0. Let L be a non negative selfadjoint operator on L2(X). We assume that e-tL satisfies a Gaussian upper bound and that the flow eitL satisfies a typical L1- L∞ dispersive estimate of the form ‖eitL‖L1→L∞≲|t|-n/2. Then we prove a similar L1- L∞ dispersive estimate for a general class of flows eitϕ(L), with φ(r) of power type near 0 and near ∞. In the case of fractional powers φ(L) = Lν, ν∈ (0 , 1) , we deduce dispersive estimates for eitLν with data in Sobolev, Besov or Hardy spaces HLp with p∈ (0 , 1] , associated to the operator L

Local well-posedness for the space-time Monopole equation in Lorenz gauge

It is known from the work of Czubak that the space-time Monopole equation is locally well-posed in the Coulomb gauge for small initial data in $H^s(\mathbb{R}^2)$ for $s>1/4$. Here we prove local well-posedness for arbitrary initial data in $H^s(\mathbb{R^2})$ with $s>1/4$ in the Lorenz gauge.Comment: To appear in NoDE

Intensidad turística en ciudades medianas y pequeñas españolas. Investigación de turismo urbano mediante un indicador cuantitativo: Indicador de Intensidad Turística

Urban tourism has experienced significant growth in recent decades, generating situations of overtourism and tourismophobia. In Spain, cities such as Barcelona, Sevilla, Valencia, and Madrid face the effects of excess tourism on a daily basis, even in times of the pandemic. The objective of this study is to determine whether an increase in tourism intensity has also occurred in small and medium-sized Spanish cities and, if so, to develop an assessment of the phenomenon. To this end, a Tourism Intensity Index (TII) has been designed and applied to international tourist flows and statistically tested (through cluster analysis and Student’s ‘t’ test), permitting not only a confirmation of the phenomenon but its characterization during the period 2003-2020 and its interpretation from a geographical perspective. This research shows that tourist intensity is a phenomenon that extends also to medium and small Spanish cities.En las últimas décadas el turismo urbano ha experimentado un importante crecimiento generando situaciones de overtourism y turismofobia. En España, ciudades como Barcelona, Palma de Mallorca o Madrid afrontan diariamente los efectos de los excesos turísticos, incluso en tiempos de pandemia. Precisamente, es objetivo de este estudio determinar si también se está produciendo un incremento de la intensidad turística en las ciudades españolas de tamaño medio y pequeño y, en caso positivo, desarrollar una valoración del fenómeno. Para ello, se ha diseñado un Indicador de Intensidad Turística (ITT) cuyos resultados, aplicados a los flujos de turismo internacional y testeados estadísticamente (análisis clúster y prueba “t”-Student), han permitido no solo confirmar el fenómeno sino también caracterizarlo durante el periodo cronológico 2003-2020 e interpretarlo desde una perspectiva geográfica. Esta investigación muestra que la intensidad turística es un fenómeno que también deben afrontar las ciudades españolas pequeñas y medianas que han experimentado en los últimos años un importante crecimiento turístico.This research forms part the competitive project “Culture and territory in Spain: Processes and impacts in small and medium-sized cities” (references CSO2017-83603-C2-1-R and CSO2017-83603-C2-2 -R), financed by the State Research Program “Development and Innovation Oriented to the Challenges of Society” of the Spanish Ministry of Economy and Competitiveness, within the framework of the State Plan for Scientific and Technical Research and Innovation, 2013-2016