[vc_section el_class="container mx-auto align-items-center circle--pattern" css=".vc_custom_1648956589196{padding-top: 3rem !important;}"][vc_row el_class="pb-5"][vc_column][vc_wp_custommenu nav_menu="6"][uoh_breadcrumb_component automatic_breadcrumb="true"][uoh_title_component title_dropdown="big" title_decorator="true"]{{title}}[/uoh_title_component][vc_column_text css=""]This research proposal aims to study the long-term behavior of solutions to partial differential equations arising from dispersive dynamics, kinetic models, and integro-differential dynamics in ecology; and to study extremals of functional inequalities in connection to the ground states of partial differential equations arising from quantum mechanics and diffusion phenomena. Five major topics are proposed: Relativistic quantum mechanics, Dirac operators and functional inequalities; Symmetry breaking in weighted functional inequalities and weighted diffusions; long time dynamics in dispersive PDEs in one space dimension; long-term dynamics in nonlocal models from ecology; and hypocoercivity and decay to equilibrium in kinetic models with heavy tails. The first topic focuses on establishing connections between spectral problems and functional inequalities for Dirac operators. The aim is to analyse the symmetry of optimal spinors in inequalities of Keller-Lieb-Thirring type, and to obtain the solitary waves of Soler-type nonlinear Dirac equations as optimizers of a nonlinear inequality. The second topic aims to characterize a symmetry range in which optimal functions are radially symmetric for weighted logarithmic Sobolev inequalities and a new family of Caffarelli-Kohn-Nirenberg inequalities. A nonlinear carré-du-champ method will be adapted to prove entropy-type estimates. Rigidity, perturbation, and stability issues will be addressed. The third topic seeks to study the asymptotic stability of topological and non-topological solitions for a class of dispersive PDEs in dimension one. A new method is proposed, based on perturbations in weighted spaces with exponential weights, on the so-called virial identities, and on the study of existence of breathers. The fourth topic concerns the description of evolutionary stable strategies of long-term dynamics of integro-differential models that arise in the modeling of structured populations, and to obtain qualitative and quantitative insights on the concentration dynamics. In the fifth topic, the aim is to extend the Dolbeault-Mouhot-Schmeiser method to study the large-time behavior of solutions for a broad family of kinetic equations in which the confinement potential exhibits heavy tails. The goals of this project are multiple: to strengthen and to create new collaborative research networks between France and Chile in the field of nonlinear partial differential equations and applications, to publish co-authored articles in top-tier journals and disseminate the results in international meetings, and to promote the formation of advanced human capital. In order to achieve these goals, yearly workshops will be organized in France and Chile to account for the progress of the investigations as well as to encourage the participation of students and young researchers. International training of doctoral and postdoctoral researchers will be ensured by allocating resources from this project for exchanges. Considering the history of successful collaboration amongst the members of this project, and their expertise in their research fields, we are confident about the successful termination of the project. In particular, we expect to pave the way for new research avenues. The main scientific contribution of this proposal involves adapting state-of-the-art techniques from PDEs and nonlinear analysis to obtain qualitative and quantitative results for variational problems and partial differential equations, in which the setting plays a crucial role: complex-valued matrices (first topic), nonlinear and weighted (second topic), strongly nonlinear and dispersive (third topic), nonlocal (fourth topic), general assumption on the tail of the confining potential (fifth topic). This proposed research will provide insights into spectral theory, stability theory of equilibria of differential equations, optimal rates of convergence to equilibria, and their relation to optimal constants in functional inequalities. The expected results will help improving the understanding of various real-life phenomena, including population-dynamics, relativistic quantum mechanics, and diffussion processes. The viability of the project is sustained on the expertise of the members of the Chilean and French research teams, including experts in partial differential equations, nonlinear analysis, calculus of variations, and mathematical physics. Their successful collaboration record and significant contributions to these fields only strengthen the potential of this proposal. In conclusion, the present research project will not only foster the scientific cooperation between Chile and France but it will also provide meaningful advancements in the aforementioned fields and their application to various physical phenomena.[/vc_column_text][/vc_column][/vc_row][/vc_section][vc_section css=".vc_custom_1649209804184{background-color: #f6faff !important;}" el_class="p-md-0 pt-md-5"][vc_row el_class="container mx-auto align-items-center p-md-0 pt-5"][vc_column el_class="p-0"][/vc_column][/vc_row][/vc_section][vc_section css=".vc_custom_1649210787516{background-color: #f6faff !important;}" el_class="p-md-0 pt-md-5 pb-md-5"][vc_row el_class="container mx-auto align-items-center"][vc_column][/vc_column][/vc_row][/vc_section]