Juan Campos

ASYMMETRIC ELLIPTIC PROBLEMS WITH INDEFINITE WEIGHTS

We prove the existence of a first nontrivial eigenvalue for an asymmetric elliptic problem with weights involving the laplacian (cf. (1.2) below) or more generally the p-laplacian (cf. (1.3) below). A first application is given to the description of the beginning of the Fuy

On the structure of the set of bounded solutions on a periodic Liénard equation

We describe the dynamics of a class of second order periodic differential equations whose main feature is a monotone nonlinearity. It is proved that the set of bounded solutions is homeomorphic to the graph of a decreasing function.

Pattern formation in a flux limited reaction–diffusion equation of porous media type

A non-linear PDE featuring flux limitation effects together with those of the porous media equation (non-linear Fokker–Planck) is presented in this paper. We analyze the balance of such diverse effects through the study of the existence and qualitative behavior of some admissible patterns, namely traveling wave solutions, to this singular reaction–diffusion equation. We show the existence and qualitative behavior of different types of traveling waves: classical profiles for wave speeds high enough, and discontinuous waves that are reminiscent of hyperbolic shock waves when the wave speed lowers below a certain threshold. Some of these solutions are of particular relevance as they provide models by which the whole solution (and not just the bulk of it, as it is the case with classical traveling waves) spreads through the medium with finite speed.

Flux-saturated porous media equations and applications

The aim of this paper is to review the main recent results about the dynamics of nonlinear partial differential equations describing flux-saturated transport mechanisms, eventually in combination with porous media flow and/or reactions terms. The result is a system characterized by the presence of wave fronts which move defining an interface. This can be used to model different process in applications in a variety of areas as developmental biology or astrophysics. The concept of solution and its properties (well-posedness in a bounded variation scenario, Rankine–Hugoniot and geometric conditions for jumps, regularity results, finite speed of propagation, . . . ), qualitative study of these fronts (traveling waves in particular) and application in morphogenesis cover the panorama of this review. J. Calvo, J. Campos, O. Sánchez and J. Soler were supported in part by MINECO (Spain), project MTM2014-53406-R, FEDER resources, and Junta de Andalucía Project P12-FQM-954. J. Calvo is also partially supported by a Juan de la Cierva grant of the Spanish MEC and La Caixa “Collaborative Mathematical Research”. Juan Campos is also supported by MICINN Grant with FEDER funds MTM2011-23652. V. Caselles was supported in part by MICINN (Spain), project MTM2009-08171, and also acknowledges the partial support by GRC reference 2009 SGR 773, and by “ICREA Acadèmia” prize for excellence in research funded both by the Generalitat de Catalunya. J. Calvo, Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Bellaterra, Barcelona, Spain E-mail: jcalvo@crm.cat J. Campos, Departamento de Matemática Aplicada, Facultad de Ciencias, Campus de Fuentenueva, Universidad de Granada, 18071 Granada, Spain E-mail: campos@ugr.es V. Caselles, Departamento de Tecnología, Universitat Pompeu-Fabra, Roc Boronat, 138, 08018 Barcelona, Spain E-mail: vicent.caselles@upf.edu O. Sánchez, Departamento de Matemática Aplicada, Facultad de Ciencias, Campus de Fuentenueva, Universidad de Granada, 18071 Granada, Spain E-mail: ossanche@ugr.es J. Soler, Departamento de Matemática Aplicada, Facultad de Ciencias, Campus de Fuentenueva, Universidad de Granada, 18071 Granada, Spain E-mail: jsoler@ugr.es 132 J. Calvo, J. Campos, V. Caselles, O. Sánchez and J. Soler Mathematics Subject Classification (2010). 35K57, 35B36, 35K67, 34Cxx, 70Kxx; 35B60, 37Dxx, 76B15, 35Q35, 37D50, 35Q99.

Maximum principles around an eigenvalue with constant eigenfunctions

A class of linear operators L + lambda I between suitable function spaces is considered, when 0 is an eigenvalue of L with constant eigenfunctions. It is proved that L + lambda I satisfies a strong maximum principle when lambda belongs to a suitable pointed left-neighborhood of 0, and satisfies a strong uniform anti-maximum principle when lambda belongs to a suitable pointed right-neighborhood of 0. Applications are given to various types of ordinary or partial differential operators with periodic or Neumann boundary conditions.

Qualitative behaviour for flux-saturated mechanisms: travelling waves, waiting time and smoothing effects

This paper is devoted to the analysis of qualitative properties of flux-saturated type operators in dimension one. Specifically, we study regularity properties and smoothing effects, discontinuous interfaces, the existence of traveling wave profiles, sub- and super-solutions and waiting time features. The aim of the paper is to better understand these kind of phenomena throughout two prototypic operators: The relativistic heat equation and the porous media flux-limited equation. As an important consequence of our results we deduce that solutions to the one-dimensional relativistic heat equation become smooth inside their support on the long time run.

Large minimal period orbits of periodic autonomous systems

We prove the existence of periodic orbits with minimal period greater than any prescribed number for a natural Lagrangian autonomous system in several variables that is analytic and periodic in each variable and whose potential is nonconstant.

Massera's theorem for monotone dynamical systems in three dimensions

In this paper we give a variant of the classical second Masseras theorem for three-dimensional periodic systems, that satisfy different types of monotonicity assumptions.

The functional Fučik spectrum has empty interior

We define a functional version of the Fucik spectrum for the Laplacian and we prove that this functional spectrum has empty interior.