Differential Equations

Time periodic solutions for 3D quasi-geostrophic model.

Event Date: Sep 27, 2022 in Differential Equations, Seminars

Abstract: The aim of this talk is to study time periodic solutions for 3D inviscid quasigeostrophic model. We show the existence of non trivial simply-connected rotating patches by suitable perturbation of stationary solutions given by generic revolution shapes around the vertical axis. The construction of those special solutions are done through bifurcation theory. In general, the spectral problem is very delicate and strongly depends on the shape of the initial stationary solutions. More specifically, the spectral study can be related to an eigenvalue problem of a self-adjoint compact...

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Long-time behavior of a sexual reproduction model under the effect of strongly convex selection.

Event Date: Sep 20, 2022 in Differential Equations, Seminars

Abstract: The Fisher infinitesimal model is a widely used statistical model in quantitative genetics that describes the propagation of a quantitative trait along generations of a population subjected to sexual reproduction. Recently, this model has pulled the attention of the mathematical community and some integro-differential equations have been proposed to study the precise dynamics of traits under the coupled effect of sexual reproduction and natural selection. Whilst some partial results have already been obtained, the complete understanding of the long-time behavior is essentially...

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On large solutions for fractional Hamilton-Jacobi equations.

Event Date: Aug 25, 2022 in Differential Equations, Seminars

Abstract: In this talk I will report some multiplicity results for large solutions of fractional Hamilton-Jacobi equations posed on a bounded domain, subject to exterior Dirichlet conditions. We construct large solutions using the method of sub and supersolutions, following the classical approach of J.M. Lasry and P.L. Lions for second-order equations with subquadratic gradient growth. We identify two classes of solutions: the one coming from the natural scaling of the problem; and a one-parameter family of solutions, different from the previous, which can be formally described as a...

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A multiple time renewal equation for neural assemblies with elapsed time model.

Event Date: Aug 23, 2022 in Differential Equations, Seminars

Abstract: We introduce and study an extension of the classical elapsed time equation in the context of neuron populations that are described by the elapsed time since last discharge. In this extension we incorporate the elapsed since the penultimate discharge and we obtain a more complex system of integro-differential equations. For this new system we prove convergence to stationary state by means of Doeblin’s theory in the case of weak non-linearities in an appropriate functional setting, inspired by the case of the classical elapsed time equation. Moreover, we present some numerical...

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On the fractional Zakharov-Kuznetsov equation.

Event Date: Aug 16, 2022 in Differential Equations, Seminars

Abstract: In this talk, we will present some new results related to the regularity properties of the initial value problem (IVP) for the equation” Bt u ́ Bx1 ( ́∆)α/2u + uBx1 u = 0, 0 ă α ă 2, u(x, 0) = u0(x), x = (x1, x2, . . . , xn ) P Rn, n ě 2, t P R, (0.1) where ( ́∆)α/2 denotes the n ́dimensional fractional Laplacian. In the particular case that α= 2, the equation is known as the Zakharov-Kuznetsov-(ZK) equation and it was proposed by Zakharov and Kuznetsov as amodel to describe the propagation of ion-sound waves in magnetic fields in dimen-sion n = 3. A property that enjoys the...

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p-harmonic functions with Neumann conditions and measure data.

Event Date: Aug 09, 2022 in Differential Equations, Seminars

Abstract: In this talk I will discuss the problem of finding solutions to some nonlinear elliptic equations with measure data. To this end I will introduce the concept of Renormalized Solutions, which is a very useful tool to solve both Dirichlet and Neumann problems. I will present some results in this area and also discuss some open problems.  

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