## Ohio State University Partial Differential Equations Seminar## Year 2017-2018Time/Location: Tuesdays 1:00 - 2:00pm / MW 154 (unless otherwise noted) |
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Abstract: This talk is concerned with the behavior of positive solutions to the logarithmic diffusion equation. It is known that due to fast diffusion of the equation, the extinction of solutions may occur in finite time. I will discuss the behavior of hot spots (maximum points) of solutions near the extinction time in the one-dimensional case. By applying the intersection number principle, it is shown that the hot spot typically converges to a point or goes to spatial infinity. However, in some cases, the hot spot remains bounded but does not converge to any particular point.

Leo Girardin, Paris VI (Sep 19, 2017)

Title: Non-cooperative Fisher-KPP systems: traveling waves and long-time behaviorAbstract: I will present recent results about a family of reactionâdiffusion systems whose prototype is the LotkaâVolterra competitive system with diffusion and mutations. These systems are nonlinear and non-cooperative, which makes their study difficult; the idea is then to notice and to use the underlying KPP structure.

Title: Conservation Laws, Shocks, and Random Initial Conditions

Abstract: Conservation laws have a wide range of applications to problems in fluid mechanics, turbulence, kinetic theory and many others. As perhaps the simplest quasilinear PDE, Burgers' equation has an important role as a prototype or test case for analysis for qualitative properties. I will discuss two related probabilistic approaches for which one can model the evolution of shocks through objects known as n-point functions, generalized analogs of a probability density for the solution at various points. In the second approach, I will illustrate how this method leads to solutions that persist through shock interactions, without need for any sort of resetting procedure, which is required for many methods.

Title: Asymptotic behavior of solutions to Hessian equations over exterior domains

Abstract: We present a unified approach to quadratic asymptote of solutions to a class of fully nonlinear elliptic equations over exterior domains, including Monge-Ampere equations (previously known), special Lagrangian equations, quadratic Hessian equations, and inverse harmonic Hessian equations. The argument is based on the consequence of our "exterior" Evans-Krylov--an exterior Liouville type result for general fully nonlinear elliptic equations toward constant asymptotics of bounded Hessian--and also certain rotation arguments developed earlier toward Hessian bound. This is joint work with Dongsheng Li and Zhisu Li.

Title: Asymptotic behavior of solutions to Hessian equations over exterior domains

Abstract: We present a unified approach to quadratic asymptote of solutions to a class of fully nonlinear elliptic equations over exterior domains, including Monge-Ampere equations (previously known), special Lagrangian equations, quadratic Hessian equations, and inverse harmonic Hessian equations. The argument is based on the consequence of our "exterior" Evans-Krylov--an exterior Liouville type result for general fully nonlinear elliptic equations toward constant asymptotics of bounded Hessian--and also certain rotation arguments developed earlier toward Hessian bound. This is joint work with Dongsheng Li and Zhisu Li.

Chris Henderson (Nov 14, 2017)

Title: A local-in-time Harnack inequality and applications to reaction-diffusion equationsAbstract: The classical Harnack inequality requires one to look back in time to relate the suprema and infima of a solution to a parabolic equation. In this talk, I will introduce a Harnack-type inequality that allows us to remove this looking-back-in-time restriction at the expense of a slightly weaker bound. I will then discuss applications of this bound to (time permitting) three non-local reaction-diffusion equations arising in biology and combustion. In particular, in each case, this inequality allows us to show that solutions to these equations, which do not enjoy a maximum principle, may be compared with solutions to a related local equation, which does enjoy a maximum principle. Precise estimates of the propagation speed follow from this.

Yanhui Zhang, Beijing Technology and Business University (Nov 21, 2017)

Title: Integral Representations of a Class of Harmonic Functions and the Asymptotic Behavior of Fractional Laplacians in the Half SpaceAbstract: In this article, motivated by the classic Hadamard factorization theorem about an entire function of finite order in the complex plane, we firstly prove that a harmonic function whose positive part satisfies some growth conditions, can be represented by its integral on the boundary of the half space. By using Nevanlinna’s representation of harmonic functions and the modified Poisson kernel of the half space, we further prove a representation formula through integration against a certain measure on the boundary hyperplane for harmonic functions not necessarily continuous on the boundary hyperplane whose positive parts satisfy weaker growing conditions than the first question. The result is further generalized by involving a parameter m dealing with the singularity at the infinity. Moreover, we will give the asymptotic behaviors of fractional Laplacians in the half space of Rn and discuss the weighted boundary limits of modified fractional Laplacians. The asymptotic behaviors hold outside an explicitly defined exceptional set, whose size can be controlled and set to be as small as required.

Gael Raoul, Ecole Polytecnhique (Nov 28, 2017)

Title: A PDE model to investigate the impact of climate change on a populationAbstract: We consider a population structured by a phenotypic trait and a spatial variable. In this talk we will only consider asexual populations. The model is then a parabolic equation with a non-local competition term. We will relate the dynamics of the solutions to the properties of the linearized equation. For simple environments, this connection will provide explicit formula for the survival and propagation speed of the population. The dynamics of populations in more complicated environments is important for biological applications: can a natural reserve help a population to survive? will a mountain stop an invasive species? etc. We will explain how the methods described above could provide an interesting insight for such problems.

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