The effect on Fisher-KPP propagation in a cylinder with fast diffusion on the boundary

In this paper we consider a reaction-diffusion equation of Fisher-KPP type inside an infinite cylindrical domain in $\mathbb{R}^{N+1}$, coupled with a reaction-diffusion equation on the boundary of the domain, where potentially fast diffusion is allowed. We will study the existence of an asymptotic speed of propagation for solutions of the Cauchy problem associated with such system, as well as the dependence of this speed on the diffusivity at the boundary and the amplitude of the cylinder. When $N=1$ the domain reduces to a strip between two straight lines. This models the effect of two roads with fast diffusion on a strip-shaped field bounded by them.Comment: 31 pages, 3 figure

Relaxation of a scalar nonlocal variational problem with a double-well potential

We consider nonlocal variational problems in Lp, like those that appear in peridynamics, where the functional object of the study is given by a double integral. It is known that convexity of the integrand implies the lower semicontinuity of the functional in the weak topology of Lp. If the integrand is not convex, a usual approach is to compute the relaxation, which is the lower semicontinuous envelope in the weak topology. In this paper we compute such a relaxation for a scalar problem with a double-well integrand. The relaxation is non-trivial, and, contrary to the local case, it cannot be represented as a double integral, as the original problem. Nonetheless, we show that, as for the local case, the relaxation can be expressed in terms of the energy of a suitable truncation of the considered functionThis work has been supported by the Spanish Ministry of Economy and Competitivity through project MTM2017-85934-C3-2-P (C.M.-C.) and project PGC2018-097104-B-100 and Juan de la Cierva Incorporation fellowship IJCI-2015-25084 (A.T.

Mathematical analysis and numerical treatment of a class of superlinear indefinite boundary value problems of elliptic type

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Matemática aplicada, leída el 18-06-2013Esta Tesis Doctoral estudia la existencia, multiplicidad y estabilidad de soluciones de un problema elíptico semilineal planteado en un dominio acotado de R^N, con un peso delante de la nolinealidad que cambia de signo y con condiciones de Dirichlet no homogéneas en su frontera. En el Capítulo 1 se presenta un resultado de multiplicidad elevada en dimensión 1 con el peso constante a trozos, obtenido con técnicas de diagramas de fases y un análisis exhaustivo de aplicaciones de Poincaré. Además se determina la estructura y se demuestran las principales propiedades cualitativas de los diagramas de bifurcación subyacentes, usando para ello la amplitud de la parte superlineal como parámetro. En el Capítulo 2 se estudia el caso general con pesos arbitrarios en cualquier dimensión. Se prueba, con métodos de continuación y la identidad de Picone, que la única solución estable del modelo es la minimal, que para valores grandes del parámetro de bifurcación no hay soluciones y, con técnicas de tipo topológico, que, en presencia de cotas a priori, salvo en el punto de retorno maximal, si hay una solución, entonces debe haber, al menos, dos. En el Capítulo 3 se presentan los diagramas de bifurcación relativos al Capítulo 1, computados numéricamente con métodos de continuación. Para obtenerlos, ha sido necesario adaptar los algoritmos de continuación existentes para solventar los problemas computacionales relacionados con la complejidad de los diagramas y sus propiedades cuantitativas. Sólo entonces ha sido posible computar numéricamente tales diagramas para pesos más generales que los del Capítulo 1, constatándose que se cumplen patrones similares de multiplicidad elevada, aunque la estructura topológica del diagrama puede variar dramáticamente. Estos resultados pueden tener importantes aplicaciones en Ecología, ya que la ecuación modela los estados estacionarios de una especie cuyos individuos compiten en unas zonas del territorio, mientras que cooperan en otras.This Thesis studies the existence, multiplicity and stability of the solutions of a nonlinear elliptic problem, posed in a bounded domain of R^N, with a weight in front of the non-linearity that changes sign and non-homogeneous Dirichlet boundary conditions. In Chapter 1 we present a result of high multiplicity in dimension 1 with a piecewise constant weight, based on phase portrait techniques and an exhaustive analysis of Poincaré maps. Moreover the structure of the underlying bifurcation diagrams is determined and their main qualitative properties are proved, using the amplitude of the super-linear part as the main bifurcation parameter. In Chapter 2 the general case with arbitrary weights in any dimension is studied. Using continuation methods and a Picone identity, we prove that the unique stable solution of the model is the minimal one. Moreover for large values of the main bifurcation parameter there are no solutions, while, with topological techniques in the presence of a priori bounds, when there is a solution, there must be at least two, apart from the maximal turning point. In Chapter 3 we present the bifurcation diagrams related to Chapter 1, which have been numerically computed via path following methods. To obtain them we had to adapt the available numerical algorithms in order to overcome some computational problems arising from the complexity of the diagrams and their quantitative properties. Once these problems have been solved, we could compute the diagrams for more general weights than those of Chapter 1, ascertaining that similar high multiplicity patterns hold, even if the topological structure of the diagram can vary dramatically. These results can have important applications in Ecology, since the equation models the steady states of a species whose individuals compete in some regions of the habitat, while they cooperate in other.Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEunpu