I will present a new PDE approach to obtain large time behavior
of Hamilton-Jacobi equations. This applies to usual Hamilton-Jacobi
equations, as well as the degenerate viscous cases, and weakly coupled
systems. The degenerate viscous case was an open problem in last 15 years.
This is the joint work with Cagnetti, Gomes, and Mitake.

We consider the aggregation equation ρt − ∇ · (ρ∇K ∗ ρ) = 0 in Rn, where the interaction potential K models short-range repulsion and long-range attraction. We study a family of interaction potentials with repulsion given by a Newtonian potential and attraction in the form of a power law. We show global well-posedness of solutions and investigate analytically and numerically the equilibria and their global stability. The equilibria have biologically relevant features, such as finite densities and compact support with sharp boundaries. This is joint work with Yanghong Huang and Theodore Kolokolnikov. 

In the talk I will describe my recent work, joint with Carlos Kenig and
Fanghua Lin, on homogenization of the Green and Neumann functions for a family of second order elliptic systems with highly oscillatory periodic coefficients. We study the asymptotic behavior of the first derivatives of the Green and Neumann functions, using Dirichlet and Neumann correctors. As a result, we obtain asymptotic expansions of Poisson kernels and the Dirichlet-to-Neumann maps as well as optimal convergence rates in L^p and W^{1,p} for solutions with Dirichlet or Neumann boundary conditions.

In this talk I will discuss the local and global well-posedness of coupled non- linear wave equations with damping and supercritical sources. Our interests lie in the interaction between source and damping terms and their influence on the behavior of solutions. I will introduce the method of using the monotone operator theory to obtain the local existence of weak solutions to our system. Also we extend a result by Brezis on convex integrals on Sobolev spaces, which allows us to overcome a major technical difficulty in the proof of the existence of solutions. 

The talk focuses on positive equilibrium (i.e. time-independent) solutions to mathematical models for the dynamics of populations structured by age and spatial position. This leads to the study of quasilinear parabolic equations with nonlocal and possibly nonlinear initial conditions. We shall see in an abstract functional analytic framework how bifurcation techniques may be combined with optimal parabolic regularity theory to establish the existence of positive solutions. As an application of these results we give a description of the geometry of coexistence states in a two-parameter predator-prey model.