Sobre la caracterización y robustez de los atractores de sistemas dinámicos multivaluados
The objective of the present thesis is studying multivalued dynamical systems. In particular, we pretend to obtain results related with the structure of the attractors in order to describe the behaviour of solutions for different equations. Therefore, our research may be situated in the field of App...
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| Tipo de recurso: | tesis doctoral |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Universidad Miguel Hernández de Elche |
| Repositorio: | REDIUMH. Depósito Digital de la UMH |
| OAI Identifier: | oai:dspace.umh.es:11000/28947 |
| Acceso en línea: | https://hdl.handle.net/11000/28947 |
| Access Level: | acceso abierto |
| Palabra clave: | Ecuaciones diferenciales Derivadas parciales CDU::5 - Ciencias puras y naturales::51 - Matemáticas |
| Sumario: | The objective of the present thesis is studying multivalued dynamical systems. In particular, we pretend to obtain results related with the structure of the attractors in order to describe the behaviour of solutions for different equations. Therefore, our research may be situated in the field of Applied Mathematics. Specifically, Chapter 1 deals with robustness of dynamically gradient multivalued semiflows. As an application, we describe the dynamical properties of a family of Chafee-Infante problems approximating a differential inclusion, proving that the weak solutions of these problems generate a dynamically gradient multivalued semiflow with respect to suitable Morse sets. Chapter 2 focus on a more general equation called nonlocal reaction-diffusion equation in which the diffusion depends on the gradient of the solution. Firstly, we prove the existence and uniqueness of regular and strong solutions. Secondly, we obtain the existence of global attractors in both situations under rather weak assumptions by defining a multivalued semiflow. We finish this section characterizing the attractor either as the unstable manifold of the set of stationary points or as the stable one when we consider solutions only in the set of bounded complete trajectories. In the last chapter we study the structure of the global attractor for the multivalued semiflow generated by a nonlocal reaction-diffusion equation in which we cannot guarantee uniqueness of the Cauchy problem. We start analysing the existence and properties of stationary points, showing that the problem undergoes the same cascade of bifurcations as in the Chafee-Infante equation. To conclude, we study the stability of the fixed points and establish that the semiflow is dynamically gradient. Also, we prove that the attractor consists of the stationary points and their heteroclinic connections and analyse some of the possible connections. Apart from these three chapters, the manuscript contains an unnumbered section, Introduction (and its Spanish version), as a preamble, where the work as well as the objetives that we pretend to cover are exposed. Subsequently, we have included the preliminary Chapter 0 in order to detail the framework and the previous results needed to achieve the proposed objectives. To end this work, we have created two unnumbered sections, Appendix A and Conclusions and future work (and its Spanish version). In the first one, details about generalization of the lap number property are given whilst in the other one main contributions of our research and some comments on future research lines are summarized. |
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