The liouville equation in L1 spaces

AbstractWe consider the first order equation ∂u∂t=a·▽u in the Banach lattice L1(RN). By requiring a minimal amount of Sobolev regularity on the vector-field α, we show that α·▿ generates a C0-group, thereby generalizing a result of [1]. From there, we conclude the well-posedness of Liouville equation ∂u∂t= -ξ·▽xu+▽xV·ξu, for a given potential V. The comparison between the general and force-free Liouville evolution yields the existence of the wave and scattering operators, which in turn are used to prove that the spectrum of the Liouville operator is purely residual in L1(R6)

DELSARTE'S EQUATION FOR CAPUTO'S OPERATORS

Delsarte's equation is investigated for Caputo's differential operators. Solv-ability of the resulting fractional hyperbolic Cauchy problem is achieved in the sense of distributions. A regularity result shows that the solution may be a function of time. Rigorous Delsarte's representations are established. The symmetry between the fractional operators acting on space and time, induced by Delsarte's equation, opens the door to new type of fractional PDE's

FEYNMAN PATH FORMULA FOR TIME FRACTIONAL SCHRÖDINGER TYPE EQUATION

In this paper for the Mittag-Leffler function E_α (z) we dene E_α (t^α A), when A is the generator of an uniformly bounded (C_0) semigroup. For the Hamiltonian H = − 2m ∆ + V (x) we express E_α (t^α H) by subordination principle of the Feynmann path integral and we retrieve the corresponding Green function

The Asymptotic Behavior of a Transport Equation in Cell Population Dynamics with a Null Maturation Velocity

AbstractIn this paper we complete the study of Rotenberg's model (M. Rotenberg, 1983, J. Theoret. Biol.103, 181–199) describing the growth of a cell population and treated partially by M. Boulanouar and H. Emamirad (Differential Integral Equations, 13 (2000), 125–144). In contrast to our previously cited treatment, here we impose the condition that the maturation velocity for any cell can become null. This consideration implies that the cell population never completely leaves its initial distribution, because at every time we can find some cells of initial cell population that are not divided. In this case, the generated semigroup is not compact. To surmount this difficulty, after studying the irreducibility of the generated semigroup, we calculate explicitly its essential type and we show the asymptotic convergence of the generated semigroup to a projection of rank 1

C-spectrality of the Schrödinger operator in Lp spaces

AbstractIn [1], the notions of C-regularized functional calculus and C-regularized scalar operator are defined and their mutual relationship with temperate C-regularized groups is given. In this note, we apply these notions in two ways: first we consider the Schrödinger operator in Lp(Ω) with Dirichlet boundary condition, when Ω is a bounded domain in Rn. The second application will be the operator −Δ + V in Lp(Rn), when V belongs to the Kato's class of potentials

Distributional chaos for the Forward and Backward Control traffic model

The interest in car-following models has increased in the last years due to its connection with vehicle-to-vehicle communications and the development of driverless cars. Some non-linear models such as the Gazes–Herman–Rothery model were already known to be chaotic. We consider the linear Forward and Backward Control traffic model for an infinite number of cars on a track. We show the existence of solutions with a chaotic behaviour by using some results of linear dynamics of C0-semigroups. In contrast, we also analyse which initial configurations lead to stable solutions. © 2015 Elsevier Inc. All rights reserved.The first three authors were supported by MTM2013-47093-P. The first author was supported by the ERC grant HEVO no. 2177691. The third author was also supported by MTM2013-47093-P and by a grant from the FPU program of MEC (AP 2010-4361).Barrachina Civera, X.; Conejero, JA.; Murillo Arcila, M.; Seoane-Sepulveda, JB. (2015). Distributional chaos for the Forward and Backward Control traffic model. Linear Algebra and its Applications. 479:202-215. https://doi.org/10.1016/j.laa.2015.04.010S20221547

Distributionally chaotic families of operators on Fréchet spaces

This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Communications on Pure and Applied Analysis (CPAA) following peer review. The definitive publisher-authenticated version Conejero, J. A., Kostić, M., Miana, P. J., & Murillo-Arcila, M. (2016). Distributionally chaotic families of operators on Fréchet spaces.Communications on Pure and Applied Analysis, 2016, vol. 15, no 5, p. 1915-1939, is available online at: http://dx.doi.org/10.3934/cpaa.2016022The existence of distributional chaos and distributional irregular vectors has been recently considered in the study of linear dynamics of operators and C-0-semigroups. In this paper we extend some previous results on both notions to sequences of operators, C-0-semigroups, C-regularized semigroups, and alpha-timesintegrated semigroups on Frechet spaces. We also add a study of rescaled distributionally chaotic C-0-semigroups. Some examples are provided to illustrate all these results.The first and fourth authors are supported in part by MEC Project MTM2010-14909, MTM2013-47093-P, and Programa de Investigacion y Desarrollo de la UPV, Ref. SP20120700. The second author is partially supported by grant 174024 of Ministry of Science and Technological Development, Republic of Serbia. The third author has been partially supported by Project MTM2013-42105-P, DGI-FEDER, of the MCYTS; Project E-64, D.G. Aragon, and Project UZCUD2014-CIE-09, Universidad de Zaragoza. The fourth author is supported by a grant of the FPU Program of Ministry of education of Spain.Conejero, JA.; Kostic, M.; Miana Sanz, PJ.; Murillo Arcila, M. (2016). Distributionally chaotic families of operators on Fréchet spaces. Communications on Pure and Applied Analysis. 15(5):1915-1939. https://doi.org/10.3934/cpaa.2016022S1915193915