Depleting the signal: Analysis of chemotaxis-consumption models—A survey

We give an overview of analytical results concerned with chemotaxis systems where the signal is absorbed. We recall results on existence and properties of solutions for the prototypical chemotaxis-consumption model and various variants and review more recent findings on its ability to support the emergence of spatial structures

p-fractional Hardy–Schrödinger–Kirchhoff systems with critical nonlinearities

Abstract This paper deals with the existence of nontrivial solutions for critical Hardy–Schrödinger–Kirchhoff systems driven by the fractional p-Laplacian operator. Existence is derived as an application of the mountain pass theorem and the Ekeland variational principle. The main features and novelty of the paper are the presence of the Hardy terms as well as critical nonlinearities

Existence and Multiplicity of Solutions of Functional Differential Equations

The first part of the memory goes through those discoveries related to Green’s functions. In order to do that, first we recall some general results concerning involutions which will help us understand their remarkable analytic and algebraic properties. Chapter 1 will deal about this subject while Chapter 2 will give a brief overview on differential equations with involutions to set the reader in the appropriate research framework. In Chapter 3 we start working on the theory of Green’s functions for functional differential equations with involutions in the most simple cases: order one problems with constant coefficients and reflection. Here we solve the problem with different boundary conditions, studying the specific characteristics which appear when considering periodic, anti-periodic, initial or arbitrary linear boundary conditions. We also apply some very well known techniques (lower and upper solutions method or Krasnosel’skiĭ’s Fixed Point Theorem, for instance) in order to further derive results. Computing explicitly the Green’s function for a problem with nonconstant coefficients is not simple, not even in the case of ordinary differential equations. We face these obstacles in Chapter 4, where we reduce a new, more general problem containing nonconstant coefficients and arbitrary differentiable involutions, to the one studied in Chapter 3. To end this part of the work, we have Chapter 5, in which we deepen in the algebraic nature of reflections and extrapolate these properties to other algebras. In this way, we do not only generalize the results of Chapter 3 to the case of -th order problems and general twopoint boundary conditions, but also solve functional differential problems in which the Hilbert transform or other adequate operators are involved. The last chapters of this part are about applying the results we have proved so far to some related problems. First, in Chapter 6, setting again the spotlight on some interesting relation between an equation with reflection and an equation with a -Laplacian, we obtain some results concerning the periodicity of solutions of that first problem with reflection. Chapter 7 moves to a more practical setting. It is of the greatest interest to have adequate computer programs in order to derive the Green’s functions obtained in Chapter 5 for, in general, the computations involved are very convoluted. Being so, we present in this chapter such an algorithm, implemented in Mathematica. The reader can find in the appendix the exact code of the program. In the second part of the Thesis we use the fixed point index to solve four different kinds of problems increasing in complexity: a problem with reflection, a problem with deviated arguments (applied to a thermostat model), a problem with nonlinear Neumann boundary conditions and a problem with functional nonlinearities in both the equation and the boundary conditions. As we will see, the particularities of each problem make it impossible to take a common approach to all of the problems studied. Still, there will be important similarities in the different cases which will lead to comparable results

Homogeneización y diferenciación de formas de ecuaciones elípticas cuasilineales

Tesis de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Análisis Matemático y Matemática Aplicada, leída el 19-12-2017Esta tesis se ha divido en dos partes de tamaños desiguales. La primera parte es la componentecentral del trabajo del candidato. Se encarga de la optimización de reactores químicosde lecho fijo, y el estudio de su efectividad, como se expondrá en los siguientes párrafos. Lasegunda parte es el resultado de la visita del candidato al Prof. Häim Brezis en el InstitutoTecnológico de Israel (Technion) en Haifa, Israel. Se entra en una pregunta concreta sobrebases óptimas en L2, que es de importancia en Tratamiento de Imágenes, y que fue formuladopor el Prof. Brezis.La primera parte de la tesis, que estudio reactores químicos, se ha dividido en 4 capítulos.Estudia un modelo establecido que tiene aplicaciones directas en Ingeniería Química, y lanoción de efectividad. Una de las mayores dificultades con la que nos enfrentamos es elhecho que, por las aplicaciones en Ingeniería Química, estamos interesados en reacciones deorden menor que uni (de tipo raíz).El primer capítulo se centra en la modelización: obtener un modelo macroscópico (homogéneo)a partir de un comportamiento microscópico prescrito. A este método se le conocecomo homogeneización. La idea es considerar partículas periódicamente repetidas, de formafija G0, a una distancia ε, y que han sido reescaladas por un factor aε . La expresión habitualde este factor es aε = C0εα, donde α ≥ 1 y C0 es una constante positiva. El objetivo esestudiar los diferentes comportamientos cuando ε →0, y ya no se consideran las partículas.Primero, los casos de partículas grandes y partículas pequeños se tratan de formas distintas.Este segundo, que ha sido el central en esta tesis, se divide en subcrítico, crítico y supercrítico.En términos generales, existe un valor α∗ tal que los comportamientos de los casos α = 1(partículas grandes), 1 α∗ (partículas supercríticos) son significativamente distintos...This thesis has been divided into two parts of different proportions. The first part is the mainwork of the candidate. It deals with the optimization of chemical reactors, and the study ofthe effectiveness, as it will explained in the next paragraphs. The second part is the result ofthe visit of the candidate to Prof. Häim Brezis at the Israel Institute of Technology (Technion)in Haifa, Israel. It deals with a particular question about optimal basis in L2 of relevance inImage Proccesing, which was raised by Prof. Brezis.The first part of the thesis, which deals with chemical reactors, has been divided intofour chapters. It studies well-established models which have direct applications in ChemicalEngineering, and the notion of “effectiveness of a chemical reactor”. One of the maindifficulties we faced is the fact that, due to the Chemical Engineering applications, we wereinterested in dealing with root-type nonlinearities. The first chapter focuses on modeling: obtaining a macroscopic (homogeneous) modelfrom a prescribed microscopic behaviour. This method is known as homogenization. Theidea is to consider periodically repeated particles of a fixed shape G0, at a distance ε, whichhave been rescaled by a factor aε . This factor is usually of the form aε =C0εα, where α ≥ 1and C0 is a positive constant. The aim is to study the different behaviours as ε →0, whenthe particles are no longer considered. It was known that depending of this factor there areusually different behaviours as ε →0. First, the case of big particles and small particles aretreated differently. The latter, which have been the main focus of this chapter, are dividedinto subcritical, critical and supercritical holes. Roughly speaking, there is a critical valueα∗ such that the behaviours α = 1 (big particles), 1 α∗ (supercritical particles) are significantly different...Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEunpu