# PDE Seminar Spring 2015

### Fridays, 4:10pm, Stevenson Center 1307

Date: **Friday, January 30, 2015**

- Speaker:
**Jozsef Farkas, University of Stirling (UK)** - Title: Steady state problems in structured population dynamics.

- Abstract: In this talk we will discuss the question of existence of non-trivial steady states of some partial differential equations, which are intended to model structured populations. We reformulate the steady state problem as an abstract eigenvalue problem coupled with a fixed-point problem. This approach allows us to formulate biologically relevant conditions for the existence of a positive steady state. We will discuss the steady state problem for models with monotone infinite dimensional nonlinearities, as well as for models with non-monotone finite dimensional nonlinearities.

Date: **Friday, February 6, 2015**

- Speaker:
**Der-Chen Chang, Georgetown University, Washington DC** - Title: Analysis on Decoupled Domains in
**C**^{n}^{ } - Abstract: A domain Ω in
**C**^{n}and its boundary ∂Ω are said to be*decoupled of finite type*if there exists sub-harmonic, non-harmonic polynomials { P_{j}}_{j=1,…,n}, with P_{j}(0) = 0, such that

Ω = { (z_{1}, …, z_{n}, z_{n+1}): Im(z_{n+1}) > P_{1}(z_{1}) + … + P_{n}(z_{n}) }. We call the integer m_{j}=2+degree( ΔP_{j}) the degree of P_{j}. The “type” of Ω is m=max{m_{1}, …, m_{n}}.

In this talk, we use the method developed by Greiner-Stein and Chang-Nagel-Stein to construct the “fundamental solution” N for the ∂ – bar Neumann problem on Ω. Click here for a full abstract.

Date: **Friday, February 20, 2015**

- Speaker:
**Ugo Gianazza, University of Pavia (Italy)** - Title: Boundary regularity for degenerate and singular parabolic equations
- Abstract: I characterise regular boundary points of the parabolic p-Laplacian in terms of a family of barriers, both when p > 2 and 1 < p < 2. By constructing suitable families of barriers, I give some simple geometric conditions that ensure the regularity of boundary points.

Date: **Friday, March 13, 2015**

- Speaker:
**Yuanzhen Shao, Vanderbilt University** - Title: Degenerate and singular elliptic operators on manifolds with singularites
- Abstract: In this talk, we will establish the theory for a class of degenerate and singular elliptic operators on manifolds with singularities. Based on this theory, we investigate several linear and nonlinear parabolic equations arising from geometric analysis and degenerate boundary value problems. Emphasis will be given to geometric flows with “bad” initial metric.

Date: **Friday, March 20, 2015**

- Speaker:
**Mathew Gluck, The University of Alabama in Huntsville** - Title: Blow-up phenomena in elliptic and parabolic models
- Abstract: Click here for an abstract.

Date: **Thursday, April 23, ****2015, **SC 1432 at 1:10pm (notice the different day, time, and location).

- Speaker:
**Mats Gyllenberg, University of Helsinki (Finland)** - Title: When is a structured population model representable by a system of ordinary differential equations?
- Abstract: Models of physiologically structured populations are usually formulated in terms of PDEs with a nonlinear feedback through the environment, or as nonlinear Volterra integral equations. In both cases the resulting dynamical system is infinite-dimensional. Although basic questions like analysing steady states and determining their local stability properties have been settled for the infinite dimensional dynamical system, many important questions remain. For instance, in what manner does the population state evolve with time and possibly reach a steady state? What is the structure of the omega-limit set? Because of the in finite dimensionality of the problem these questions are hard to answer in general. In contrast, there is a highly developed qualitative theory for systems of ordinary di erential equations, where such questions can be treated. There are also highly efficient packages for solving systems of ODEs numerically, whereas corresponding methods for general structured population models are rare.Because of the arguments mentioned above, it is important to find necessary and sufficient conditions for solutions of PDEs and/or Volterra equations to be representable in terms of solutions of a system of ordinary di erential equations. This is the main purpose of my talk.

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