On the fractional Fisher information with applications to a hyperbolic-parabolic system of chemotaxis

We introduce new lower bounds for the fractional Fisher information. Equipped with these bounds we study a hyperbolic-parabolic model of chemotaxis and prove the global existence of solutions in certain dissipation regimes

On a drift-diffusion system for semiconductor devices

In this note we study a fractional Poisson-Nernst-Planck equation modeling a semiconductor device. We prove several decay estimates for the Lebesgue and Sobolev norms in one, two and three dimensions. We also provide the first term of the asymptotic expansion as $t\rightarrow\infty$.Comment: to appear in Annales Henri Poincar\'

Global solutions for a hyperbolic-parabolic system of chemotaxis

We study a hyperbolic-parabolic model of chemotaxis in dimensions one and two. In particular, we prove the global existence of classical solutions in certain dissipation regimes

Critical Keller-Segel meets Burgers on S1{\mathbb S}^1: large-time smooth solutions

We show that solutions to the parabolic-elliptic Keller-Segel system on ${\mathbb S}^1$ with critical fractional diffusion $(-\Delta)^\frac{1}{2}$ remain smooth for any initial data and any positive time. This disproves, at least in the periodic setting, the large-data-blowup conjecture by Bournaveas and Calvez. As a tool, we show smoothness of solutions to a modified critical Burgers equation via a generalization of the method of moduli of continuity by Kiselev, Nazarov and Shterenberg. over a setting where the considered equation has no scaling. This auxiliary result may be interesting by itself. Finally, we study the asymptotic behavior of global solutions, improving the existing results.Comment: 17 page

An aggregation equation with a nonlocal flux

In this paper we study an aggregation equation with a general nonlocal flux. We study the local well-posedness and some conditions ensuring global existence. We are also interested in the differences arising when the nonlinearity in the flux changes. Thus, we perform some numerics corresponding to different convexities for the nonlinearity in the equation

Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations

In this paper we consider a $d$-dimensional ($d=1,2$) parabolic-elliptic Keller-Segel equation with a logistic forcing and a fractional diffusion of order $\alpha \in (0,2)$. We prove uniform in time boundedness of its solution in the supercritical range $\alpha>d\left(1-c\right)$, where $c$ is an explicit constant depending on parameters of our problem. Furthermore, we establish sufficient conditions for $\|u(t)-u_\infty\|_{L^\infty}\rightarrow0$, where $u_\infty\equiv 1$ is the only nontrivial homogeneous solution. Finally, we provide a uniqueness result

On a generalized doubly parabolic Keller-Segel system in one spatial dimension

We study a doubly parabolic Keller-Segel system in one spatial dimension, with diffusions given by fractional laplacians. We obtain several local and global well-posedness results for the subcritical and critical cases (for the latter we need certain smallness assumptions). We also study dynamical properties of the system with added logistic term. Then, this model exhibits a spatio-temporal chaotic behavior, where a number of peaks emerge. In particular, we prove the existence of an attractor and provide an upper bound on the number of peaks that the solution may develop. Finally, we perform a numerical analysis suggesting that there is a finite time blow up if the diffusion is weak enough, even in presence of a damping logistic term. Our results generalize on one hand the results for local diffusions, on the other the results for the parabolic-elliptic fractional case

Global solutions for a supercritical drift-diffusion equation

We study the global existence of solutions to a one-dimensional drift-diffusion equation with logistic term, generalizing the classical parabolic-elliptic Keller-Segel aggregation equation arising in mathematical biology. In particular, we prove that there exists a global weak solution, if the order of the fractional diffusion $\alpha \in (1-c_1, 2]$, where $c_1>0$ is an explicit constant depending on the physical parameters present in the problem (chemosensitivity and strength of logistic damping). Furthermore, in the range $1-c_2<\alpha\leq 2$ with $0<c_2<c_1$, the solution is globally smooth. Let us emphasize that when $\alpha<1$, the diffusion is in the supercritical regime

A nonlocal model describing tumor angiogenesis

In this paper, we derive and study a new mathematical model that describes the onset of angiogenesis. This new model takes the form of a nonlocal Burgers equation with both diffusive and dispersive terms. For a particular value of the parameters, the equation reduces to ∂tp −1/2(−Δ)(α−1)/2H∂tp = −1/2(−Δ)α/2p + p∂xp − ∂xp, where H denotes the Hilbert transform. In addition to the derivation of the new model, the main novelty of the present paper is that we also prove a number of well-posedness results. Finally, some preliminary numerical results are shown. These numerical results suggest that the dynamics of the equation is rich enough to have solutions that blow up in finite time.R.G-B was supported by the project “Mathematical Analysis of Fluids and Applications”, Spain Grant PID2019-109348GA-I00 funded by MCIN/AEI/, Spain 10.13039/501100011033 and acronym “MAFyA”. This publication is part of the project PID2019-109348GA-I00/AEI/10.13039/501100011033. R.G-B is also supported by a 2021 Leonardo Grant for Researchers and Cultural Creators, BBVA Foundation, Spain. The BBVA Foundation accepts no responsibility for the opinions, statements, and contents included in the project and/or the results thereof, which are entirely the responsibility of the authors