Neste trabalho estudamos a dinâmica assintótica não linear de algumas equações parabólicas do tipo reação-difusão sob perturbações nos parâmetros e perturbações singulares no domínio do tipo dumbbell. Mais precisamente, trataremos dos atratores provenientes destes problemas, buscaremos compreender a dependência destes conjuntos assintóticos de estados em relação ao parâmetro, investigando a continuidade com taxa dos mesmos. O programa que executaremos para obtenção da taxa de continuidade dos atratores, bem como de toda a estrutura, mostra-nos fortes propriedades de dissipatividade exponencial de alguns semigrupos / In this work we study the asymptotic nonlinear dynamical of some reaction-diffusion parabolic equations under perturbations in parameter and singular perturbations in a dumbbell domain. More precisely, we treat of the attractors from these problems, we seek understand the dependence these asymptotic set of states in relationship the parameter, investigating continuity with rate. The program that we will follow to prove the continuity of the attractors with rate well as the entire structure, we show that these semigroups possess strong exponential dissipative properties

Neste trabalho estudamos a dinâmica assintótica não linear de algumas equações parabólicas do tipo reação-difusão sob perturbações nos parâmetros e perturbações singulares no domínio do tipo dumbbell. Mais precisamente, trataremos dos atratores provenientes destes problemas, buscaremos compreender a dependência destes conjuntos assintóticos de estados em relação ao parâmetro, investigando a continuidade com taxa dos mesmos. O programa que executaremos para obtenção da taxa de continuidade dos atratores, bem como de toda a estrutura, mostra-nos fortes propriedades de dissipatividade exponencial de alguns semigrupos / In this work we study the asymptotic nonlinear dynamical of some reaction-diffusion parabolic equations under perturbations in parameter and singular perturbations in a dumbbell domain. More precisely, we treat of the attractors from these problems, we seek understand the dependence these asymptotic set of states in relationship the parameter, investigating continuity with rate. The program that we will follow to prove the continuity of the attractors with rate well as the entire structure, we show that these semigroups possess strong exponential dissipative properties

Nous analysons trois modèles non-locaux décrivant la dynamique évolutive d’un trait phénotypique continu soumis à l’action conjointe des mutations et de la sélection. Nous établissons l’existence et l’unicité des solutions du problème de Cauchy, et donnons la description du comportement en temps long de la solution. Dans le premier travail nous étudions l’équation du réplicateur-mutateur en domaine non borné et généralisons aux cas des valeurs sélectives confinantes les résultats connus dans le cas harmonique. À savoir, l’existence d’une unique solution globale, régulière, convergeant en temps long vers un profil universel ; pour cela, nous employons des techniques de décomposition spectrale d’opérateurs de Schrödinger. Le deuxième travail traite d’un modèle dont la valeur sélective est densité-dépendante. Afin de montrer le caractère bien posé de l’équation, nous combinons deux approches. La première est basée sur l’étude de la fonction génératrice des cumulants, satisfaisant une équation de transport non locale et permettant d’obtenir implicitement le trait moyen. La deuxième exploite un changement de variable (formule d’Avron-Herbst), permettant d’écrire la solution en termes du trait moyen et de la solution de l’équation de la chaleur avec même donnée initiale. Finalement, nous étudions un modèle dont le taux de mutation est proportionnel à la valeur moyenne du trait. Nous établissons un lien bijectif entre ce dernier modèle et le deuxième, permettant ainsi de décrire finement la dynamique de la solution. Nous montrons en particulier la croissance exponentielle du trait moyen. / We analyze three non-local models describing the evolutionary dynamics of a continuous phenotypic trait undergoing the joint action of mutations and selection. We establish the existence and uniqueness of the solutions to the Cauchy problem, and give a description of the long-time behaviour of the solution. In the first work we study the replicator-mutator equation in the unbounded domain and generalize to cases of selective values confining the known results in the harmonic case. Namely, the existence of a unique global regular solution, converging towards a universal profile; for this, we use spectral decomposition techniques of Schrödinger operators. In the second work, we discuss a model whose fitness value is density-dependent. In order to show the well-posedness of the equation, we combine two approaches. The first is based on the study of the cumulant generating functions, satisfying a non-local transport equation and making it possible to implicitly obtain the average trait. The second uses a change of variable (Avron-Herbst formula), allowing the solution to be written in terms of the average trait and the solution of the heat equation with the same initial data. Finally, we study a model whose mutation rate is proportional to the average value of the trait. We establish a bijective link between this last model and the second, thus making it possible to describe the dynamics of the solution in detail. In particular, we show the exponential growth of the average trait.