Differential Equations

Orbital (in)stability of periodic wave solutions for phi^{4n}-models.

Event Date: Sep 29, 2020 in Differential Equations, Seminars

Abstract: In this talk we shall discuss the orbital stability/instability of periodic wave solutions to the general \phi^{4n}-models, for all n\in\N. These models are (physically meaningful) generalizations of the classical phi4 model in quantum field theory. In the case n=1, we shall see that several different explicit solutions can be obtained by direct computations. However, for n>1 periodic solutions are no longer explicit. Thus, for the general case (n>1), due to the lack of explicit formulas, together with the complexity in dealing with the nonlinearity, in order to prove that we...

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Un par de modelos epidemiológicos para la COVID-19.

Event Date: Sep 22, 2020 in Differential Equations, Seminars

Abstract: En esta charla voy a hablar sobre modelos epidemiológicos y las dificultades que un físico se puede encontrar cuando aborda nuevos campos. En concreto, veremos dos modelos diferentes en los que, además, se considera una variante que incluye dos grupos de edad diferentes. Estos modelos se aplican a diferentes regiones y/o países, como la comunidad autónoma de Andalucía en España, México, Grecia o Portugal. También hablaré sobre problemas como la identificabilidad de los parámetros y el cálculo de intervalos de confianza.

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A new solution to $\Delta u+u^p-u=0$ on the entire space.

Event Date: Sep 08, 2020 in Differential Equations, Seminars

Abstract: In this talk we develop some techniques to construct solutions of certain semilinear elliptic equations which are periodic in some variables, decaying in others, and quasiperiodic in one variable. These solutions, which are found near ground states of a lower-dimensional problem, are constructed using spatial dynamics and results from the KAM theory. The use of a suitable KAM-type theorem provides hypotheses for homogeneous equations which rely on simple scaling arguments, applying in particular to $\Delta u+u^p-u=0$ with $p>1$ Sobolev subcritical.

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Kac’s model with thermostats and rescaling.

Event Date: Sep 01, 2020 in Differential Equations, Seminars

Abstract: In this introductory talk we present Kac’s model in statistical mechanics that involves N identical particles undergoing collisions. Kac introduced this model in 1956 to derive the Kac-Boltzmann equation: a one particle equation. Kac’s approach in obtaining this equation is now known as ”propagation of chaos”. We also introduce thermostats and see their role in speeding up approach to equilibrium. Finally, we introduce a rescaling mechanism for the thermostated Kac model, and establish uniform in time propagation of chaos (with explicit rates) for this...

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Un fenómeno de concentración de soluciones para la ecuación de Yamabe en variedades producto.

Event Date: Aug 25, 2020 in Differential Equations, Seminars

Abstract: En esta charla estudiaremos multiplicidad de soluciones positivas para la ecuaci ́on de Yamabe en una variedad riemanniana producto (M × X,g + ε2h), donde (Mn,g) y (Xm,h) son variedades Riemannianas cerradas, con n ≥ 3. Emplearemos el procedimiento de reduccio ́n de Lyapunov-Schmidt para encontrar soluciones que tengan varios picos que se concentran en un punto cr ́ıtico de la curvatura es- calar de M cuando ε → 0. Este es un trabajo realizado en conjunto con Miguel Ruiz (UNAM).

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Some qualitative properties of Lane-Emden type systems.

Event Date: Aug 18, 2020 in Differential Equations, Seminars

Abstract: In this talk we discuss some symmetry and unique continuation results for systems of elliptic partial differential equations of Lane-Emden type. As an application, we obtain uniqueness of positive solutions in the subcritical regime, and nonexistence of nontrivial radial solutions in the critical and supercritical cases. Some of our results hold for Pucci’s fully nonlinear operators.  

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