arXiv subjects

Cristina Brändle

Publications and source records attributed to Cristina Brändle.

13 recordsLinked to original sources

arXiv2025

Asymmetric Kedem-Katchalsky boundary conditions for systems with spatial heterogeneities

This work investigates a model describing the interaction of two species in habiting separate but adjacent areas. These populations are governed by a system of equations that account for spatial variations in growth rates and the effects of crowding. A key feature is the presence of areas within each domain where resources are unlimited and crowding effects are absent. The species interact solely through a common bound ary interface, which is modeled by asymmetric Kedem-Katchalsky boundary conditions. The paper provides existence, non-existence, and behavior of positive solutions for the system. It is shown that a unique positive population distribution exists when one of the growth rate parameters falls within a specific range defined by two critical values. One of these critical values represents a bifurcation point where the population can emerge from extinction, while the other is determined by the characteristics of the refuge areas. The study also examines how the populations behave as the growth parameter approaches the upper critical value. This analysis reveals the phenomenon of non-simultaneous blow-up, where one population component can grow infinitely large within its refuge zone while the other remains bounded.

math.AP↗

arXiv2024

A three population Lotka-Volterra competition model with two populations interacting through an interface

In this work we consider three species competing with each other in the same habitat. One of the species lives in the entire habitat, competing with the other two species, while the other two inhabit two disjoint regions of the habitat. These two populations just interact on a region/interface which acts as a geographical barrier. This barrier condition causes a drastic change in species behaviour compared to the classical Lotka-Volterra competitive model, showing very rich and new different situations depending on the several parameters involved in the system.

math.AP↗

arXiv2024

Interface logistic problems: large diffusion and singular perturbation results

In this work we consider an interface logistic problem where two populations live in two different regions, separated by a membrane or interface where it happens an interchange of flux. Thus, the two populations only interact or are coupled through such a membrane where we impose the so-called Kedem-Katchalsky boundary conditions. For this particular scenario we analyze the existence and uniqueness of positive solutions depending on the parameters involve in the system, obtaining interesting results where one can see for the first time the effect of the membrane under such boundary conditions. To do so, we first ascertain the asymptotic behaviour of several linear and nonlinear problems for which we include a diffusion coefficient and analyse the behaviour of the solutions when such a diffusion parameter goes to zero or infinity. Despite their own interest, since these asymptotic results have never been studied before, they will be crucial in analyzing the existence and uniqueness for the main interface logistic problems under analysis. Finally, we apply such an asymptotic analysis to characterize the existence of solutions in terms of the growth rate of the populations, when both populations possess the same growth rate and, also, when they depend on different parameters.

math.AP↗

arXiv2024

A stationary population model with an interior interface-type boundary

We propose a stationary system that might be regarded as a migration model of some population abandoning their original place of abode and becoming part of another population, once they reach the interface boundary. To do so, we show a model where each population follows a logistic equation in their own environment while assuming spatial heterogeneities. Moreover, both populations are coupled through the common boundary, which acts as a permeable membrane on which their flow moves in and out. The main goal we face in this work will be to describe the precise interplay between the stationary solutions with respect to the parameters involved in the problem, in particular the growth rate of the populations and the coupling parameter involved on the boundary where the interchange of flux is taking place.

math.AP↗

arXiv2023

On an n-dimensional fourth-order system under a parametric condition

We establish the existence of positive solutions for a system of coupled fourth-order partial differential equations on a bounded domain $Ω\subset \mathbb{R}^n$\begin{align*} \left\{\begin{array}{l} Δ^2u_1 +β_1 Δu_1-α_1 u_1=f_1({ x},u_1,u_2),\\Δ^2 u_2+β_2Δu_2-α_2 u_2=f_2({ x},u_1,u_2), \end{array} \quad \quad x\inΩ, \right. \end{align*}subject to homogeneous Navier boundary conditions, where the functions $f_1,f_2 : Ω\times [0,\infty)\times [0,\infty) \rightarrow [0,\infty)$ are continuous, and $α_1,α_2,β_1$ and $β_2$ are real parameters satisfying certain constraints related to the eigenvalues of the associated Laplace operator.

math.AP↗

arXiv2018

On unbounded solutions of ergodic problems for non-local Hamilton-Jacobi equations

We study an ergodic problem associated to a non-local Hamilton-Jacobi equation defined on the whole space $λ-\mathcal{L}[u](x)+|Du(x)|^m=f(x)$ and determine whether (unbounded) solutions exist or not. We prove that there is a threshold growth of the function $f$, that separates existence and non-existence of solutions, a {phenomenon} that does not appear in the local version of the problem. Moreover, we show that there exists a critical ergodic constant, $λ_*$, such that the ergodic problem has solutions for $λ\leq λ_*$ and such that the only solution bounded from below, which is unique up to an additive constant, is the one associated to $λ_*$.

math.AP↗

arXiv2015

Nonlocal heat equations: decay estimates and Nash inequalities

We obtain $L^q$--$L^p$ decay estimates, $1\le q<p<\infty$ for solutions of nonlocal heat equations of the form $\partial_tu+\mathcal{L} u=0$. Here $\mathcal{L}$ is an integral operator given by a symmetric nonnegative kernel of Lévy type. We obtain these estimates in terms only of the behaviour of the kernel at infinity, without any information of its behaviour at the origin. This includes bounded and unbounded transition probability densities. An equivalence between the decay and a restricted Nash inequality is shown. We also prove that $\lim_{t\to \infty}\|u(t)\|_\infty=0$. Finally we deal with nonlinear nonlocal equations of porous medium type $\partial_tu+\mathcal{L}φ(u)=0$.

math.AP↗

arXiv2014

More network science for teenagers

Recently, Harrington et al.\ (2013) presented an outreach effort to introduce school students to network science and explain why researchers who study networks should be involved in such outreach activities. Based on the modules they designed and their comments on the success and failures of the activity, we have carried out a sequel with students from a high school in Madrid, Spain. We report on how we developed it and the changes we made to the original material.

physics.ed-ph↗

arXiv2011

A nonlocal one-phase Stefan problem that develops mushy regions

We study a nonlocal version of the one-phase Stefan problem which develops mushy regions, even if they were not present initially, a model which can be of interest at the mesoscopic scale. The equation involves a convolution with a compactly supported kernel. The created mushy regions have the size of the support of this kernel. If the kernel is suitably rescaled, such regions disappear and the solution converges to the solution of the usual local version of the one-phase Stefan problem. We prove that the model is well posed, and give several qualitative properties. In particular, the long-time behavior is identified by means of a nonlocal mesa solving an obstacle problem.

math.AP↗

arXiv2010

A concave-convex elliptic problem involving the fractional Laplacian

We study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the Laplacian operator together with a concave-convex term. We characterize completely the range of parameters for which solutions of the problem exist and prove a multiplicity result.

math.AP↗

arXiv2010

Unbounded solutions of the nonlocal heat equation

We consider the Cauchy problem posed in the whole space for the following nonlocal heat equation: u_t = J * u - u, where J is a symmetric continuous probability density. Depending on the tail of J, we give a rather complete picture of the problem in optimal classes of data by: (i) estimating the initial trace of (possibly unbounded) solutions; (ii) showing existence and uniqueness results in a suitable class; (iii) giving explicit unbounded polynomial solutions.

math.AP↗

arXiv2009

Large Deviations estimates for some non-local equations. General bounds and applications

Large deviation estimates for the following linear parabolic equation are studied: \[ \frac{\partial u}{\partial t}=\tr\Big(a(x)D^2u\Big) + b(x)\cdot D u + \int_{\R^N} \Big\{(u(x+y)-u(x)-(D u(x)\cdot y)\ind{|y|<1}(y)\Big\}\dμ(y), \] where $μ$ is a Lévy measure (which may be singular at the origin). Assuming only that some negative exponential integrates with respect to the tail of $μ$, it is shown that given an initial data, solutions defined in a bounded domain converge exponentially fast to the solution of the problem defined in the whole space. The exact rate, which depends strongly on the decay of $μ$ at infinity, is also estimated.

math.AP↗

arXiv2008

Large Deviations estimates for some non-local equations I. Fast decaying kernels and explicit bounds

We study large deviations for some non-local parabolic type equations. We show that, under some assumptions on the non-local term, problems defined in a bounded domain converge with an exponential rate to the solution of the problem defined in the whole space. We compute this rate in different examples, with different kernels defining the non-local term, and it turns out that the estimate of convergence depends strongly on the decay at infinity of that kernel.

math.AP↗