On the asymptotic shape of solutions to Neumann problems for non-cooperative parabolic systems

We consider a class of nonautonomous parabolic competition-diffusion systems on bounded radial domains under Neumann boundary conditions. We show that, if the initial profiles satisfy a reflection inequality with respect to a hyperplane, then global positive solutions are asymptotically (in time) foliated Schwarz symmetric with respect to antipodal points. Additionally, a related result for (positive and sign changing solutions) of a scalar equation with Neumann boundary conditions is given. The asymptotic shape of solutions to cooperative systems is also discussed.Comment: 30 pages. Revised versio

Positive powers of the Laplacian: from hypersingular integrals to boundary value problems

Any positive power of the Laplacian is related via its Fourier symbol to a hypersingular integral with finite differences. We show how this yields a pointwise evaluation which is more flexible than other notions used so far in the literature for powers larger than 1; in particular, this evaluation can be applied to more general boundary value problems and we exhibit explicit examples. We also provide a natural variational framework and, using an asymptotic analysis, we prove how these hypersingular integrals reduce to polyharmonic operators in some cases. Our presentation aims to be as self-contained as possible and relies on elementary pointwise calculations and known identities for special functions.Comment: 26 page

Qualitative properties of solutions to mixed-diffusion bistable equations

We consider a fourth-order extension of the Allen-Cahn model with mixed-diffusion and Navier boundary conditions. Using variational and bifurcation methods, we prove results on existence, uniqueness, positivity, stability, a priori estimates, and symmetry of solutions. As an application, we construct a nontrivial bounded saddle solution in the plane.Comment: New version with minor change

The phenomenological sense of poetic language in the second Heidegger

Este trabajo busca mostrar que el peculiar estilo poético que Heidegger emplea en sus obras ontohistóricas no es arbitrario ni responde a un mero afán de excentricidad. Al contrario, el filósofo vio en el lenguaje poético una manera de ser más fiel a su concepción de la fenomenología. La tesis propuesta consiste en que la fenomenología practicada en Sein und Zeit no alcanzó a dar cuenta del estado de arrojado (Geworfenheit) del Dasein ni de la esencia de la verdad entendida como rehúso. Para ello sería necesaria una fenomenología igualmente arrojada, acontecida, que encuentra su modelo rector en la poesía.This paper seeks to show that the odd poetic style used by Heidegger within his ontohistorical works is not arbitrary, nor responds to a mere desire of eccentricity. Unlike this, the philosopher found poetic language as a way to be more faithful to his conception of phenomenology. The proposed thesis is that the phenomenology practiced in Sein und Zeit failed to account for the condition of thrown (Geworfenheit) of Dasein and for the essence of truth, understood as rejection. To do so, it would be necessary a phenomenology that is equally thrown, happened, which finds its guiding model in poetry

It’s the academia, dummy! or when quantity supersedes quality

Many of us, faculty members dedicated to the profession of university professor, began our professional careers investing the best years of our youth in academically preparing ourselves in the best way possible. We continued our careers by searching for a renowned institution to work for. Its prestige provided us with a solid foundation for quick growth, thus reaching the top of the academic hierarchy, making us feel that we made the best investment for our future. Since it is our responsibility to prepare ourselves, it is also our responsibility to make sure this preparation bears the best fruit for society and our families. But also, we aspire to heighten our prestige, even a little, so that at the dusk of our professional life we can feel the satisfaction of having accomplished our mission in society and be rewarded with the appreciation and respect of our colleagues (which results in sincere affection), who we also respect and admire, and fulfill the debt to our family who supported us and sacrificed so much for our eagerness to pursue professional success, a success that has only one noble goal: academic excellence