Propagation phenomena and nonlocal equations – NONLOCAL

We will especially use the recent results obtained by the members of the project and the new perspectives that have emerged recently. We want to understand the dynamical properties of the solutions of nonlocal reaction-diffusion equations and the propagation phenomena in unbounded domains. In particular, the notions of generalized waves that have been introduced by the members of the project are the good framework to deal with other models, involving nonlocal phenomena. To achieve the main objectives presented in task 1, the analysis of related stationary problems is necessary, and it has close connections to more geometrical objects such as of nonlocal minimal surfaces. Special attention will also be put on qualitative a priori estimates, and on Harnack and Poincaré type inequalities for nonlocal equations. Another fundamental tool that is needed for the dynamical problems is the analysis of related eigenvalue problems, whose theory and applications in the nonlocal case will be dealt with in task 2. The third task is concerned with the analysis of related nonlocal models in ecology. Such models appear as excellent prototypes of nonlocal models and the notions and tools developed in the first two tasks will help in particular in the establishment of new spreading estimates for ecological invasions in heterogeneous media with long-distance dispersal or competition.

Phénomènes de propagation : blocage et propagation de fronts bistables dans des cylindres à section variable, fronts de transition généralisés dans des milieux hétérogènes, fronts pour des équations cinétiques de type Fisher-KPP, effet d'une hétérogénéité en espace pour un système champ-route, instabilité linéaire des ondes progressives d’un système thermo-diffusif en combustion, fronts et accélération dans les équations de réaction-diffusion non locales pour certaines réactions, non-existence de fronts de transition et phénomènes d’aplatissement pour équations d'évolution intégro-différentielles, effet du noyau de convolution sur la propagation de la solution.
Équations avec diffusion intégrale, propriétés fondamentales, valeurs propres, équations géométriques : théorie spectrale d'opérateurs intégro-différentiels, valeurs propres principales généralisées pour les opérateurs non locaux dans des domaines non bornés, symétrie unidimensionnelle pour des équations non locales, applications harmoniques fractionnaires du point de vue elliptique ou parabolique, espaces de Sobolev fractionnaires, solutions stationnaires de problèmes non locaux, existence et stabilité orbitale pour des systèmes de Schrödinger.
Invasions non locales en écologie mathématique : phénomènes de concentration dans certains modèles démo-génétiques, dynamique interne de composantes neutres le long de fronts d'invasions biologiques, effet d'un changement climatique sur la biodiversité, étude de la survie d'une espèce sous l'influence à la fois d'un changement climatique et d'un effet Allee, équations non locales modélisant la propagation d’épidémies dans un réseau complexe, dynamiques adaptatives et EDP non locales vérifiées par les fonctions génératrices des moments de processus stochastiques du type Wright-Fisher, croissance in vitro d'une tumeur hétérogène et problème de contrôle optimal qui en découle pour limiter la croissance.

Further analysis of propagation phenomena and the dynamics of equations with nonlocal dispersal or competition. Study of PDEs with integral diffusion, of the properties of nonlocal minimal surfaces. Analysis of nonlocal invasion problems in mathematical ecology and population genetics.

89 publications in international journals or books (April 2016).
56 talks or courses in conferences or schools (April 2016)

The objective of this research program is to understand, in the largest possible generality, propagation phenomena in nonlocal reaction-diffusion equations, e.g. equations having integral diffusions, integral velocities, memory effects, or long range interactions in the source terms. We expect very different behaviors from the known models with local diffusion, with completely new interplays between the diffusions and the nonlinearities. To our knowledge, such a program has not yet been taken up at this level of generality.

It will require, among other things, the study of fundamental linear and nonlinear PDEs with integral diffusions, as well as some incursions into the theory of nonlocal geometric movements and the development of probabilistic theoretical tools. We wish to apply our results to the treatment of propagation models in mathematical ecology, such as biological invasions or survival of a species in a changing environment. Such models are encountering a growing interest. One of the main challenges in theoretical ecology is to take into account inhomogeneities, a type of problem where numerical or asymptotic methods quickly meet their limits. This makes the approach by mathematical methods of such models especially appropriate.

Thanks to another ANR project, PREFERED, which ended in 2012 and which included 40% of the members of the NONLOCAL project as well as other scientists, the proposing team has achieved notable advances in the understanding of propagation phenomena mostly in local reaction-diffusion equations. It has established itself as a leader in the field and it has acquired an expertise that is possibly unique by its breadth. As a consequence, it appeared natural to the proposing team to apply for an ANR support to achieve a leap forward of the knowledge in the field of nonlocal equations. Furthermore, it will allow our team to enhance its visibility at national and international levels and to develop international collaborations. The achievements obtained in the last years have provided a corpus of ideas and methods that seem especially appropriate to attack nonlocal models. Although we know that we are going to face new difficulties, we are confident that the knowledge we have accumulated will be a good basis for our planned investigations. A novel feature of the NONLOCAL project is the inclusion of some probabilists, whose expertise will be beneficial in several important aspects of the scientific program. In fact, the NONLOCAL project includes a reasonable number of junior and senior scientists, and the members of the NONLOCAL project are at the leading edge of several fields of applied analysis, nonlinear PDEs, probability, as well as in mathematical ecology. These elements, as well as the unexpected phenomena related to nonlocal reaction-diffusion equations (non-standard spreading speeds, non-standard non-uniqueness and non-monotonicity results...) that they have already encountered, are highly promising for the success of the project.