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\subsection{Resumen de tesis}
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\noindent\textbf{Título}: \textit{Análisis teórico y numérico de problemas diferenciales con quimiotaxis repulsiva}.
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\textbf{Doctorando}: Diego Armando Rueda Gómez.
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\textbf{Directores}: Francisco M. Guill\'en Gonz\'alez y Mar\'{\i}a \'Angeles Rodr\'{\i}guez Bellido. Departamento de de Ecuaciones Diferenciales y An\'alisis Num\'erico e Instituto de Matem\'aticas-IMUS.
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\textbf{Centro}: Universidad de Sevilla.
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\textbf{Defensa}: 29 de octubre de 2018.
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\textbf{Calificación}: Sobresaliente cum Laude.
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This PhD thesis falls within the scopes of Theoretical and Numerical analysis of Partial Differential Equations, with applications to other sciences. Specifically, it addresses the study of some differential problems of repulsive-productive chemotaxis. The first three chapters are devoted to study a chemo-repulsion model with quadratic production, and other two chapters are focused on models with linear and potential (with a superlinear and subquadratic power) production.
In Chapter 1, we present two unconditionally mass-conservative and energy-stable time-discrete numerical schemes for a chemo-repulsion model with quadratic production, and study some additional properties of the schemes such as positivity, solvability, convergence towards weak solutions and error estimates of these schemes.
In Chapter 2, we study an unconditionally mass-conservative and energy-stable fully discrete FE scheme associated to the problem studied in Chapter 1, in which an auxiliary variable is introduced. Again, we study some properties like solvability, convergence towards weak solutions, error estimates, and weak, strong and more regular a priori estimates of the scheme. Additionally, as the scheme is nonlinear, we propose two different linear iterative methods to approach the solutions and we prove solvability and convergence of both methods to the nonlinear scheme.
In Chapter 3, we focus on the study of the asymptotic behaviour of the solutions of the model studied in Chapters 1 and 2. In the first part, we analyze the large-time behavior of the global weak-strong solutions and we prove the exponential convergence to a constant state as time goes to infinity; and in the second part, we study this same behaviour for two fully discrete FE numerical schemes associated to this model.
Finally, in Chapters 4 and 5 we focus on the study of chemo-repulsion models with linear and potential (superlinear and subquadratic) production, respectively. Here, by \nobreak{using} a regula\-ri\-za\-tion technique, we propose some unconditionally energy-stable and mass-conservative fully discrete FE schemes associated to these models, and we prove some additional properties such as solvability and approximated positivity of the solutions.
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