# PDE Seminar Fall 2017

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

Date: **Friday, September 8, 2017.**

- Speaker:
**Wandi Ding, Middle Tennessee State University.** - Title: Optimal Control applied to Elliptic and Parabolic PDEs with Biological Applications.
- Abstract: We apply the optimal control theory to Elliptic and Parabolic partial differential equations. We study the control problem of maximizing the net benefit in the conservation of a single species with a fixed amount of resources. We also consider an optimal control problem of a system of parabolic partial differential equations modelling the competition between an invasive and a native species. The existence of an optimal control is established and the uniqueness and characterization of the optimal control are investigated for both applications. Numerical simulations illustrate several cases. Some open problems are discussed.

Date: **Friday, September 15, 2017.**

- Speaker:
**Nemanja Kosovalic, University of South Alabama.** - Title: Self-Excited Periodic and Quasi-Periodic Vibrations for Higher

Dimensional Damped Wave Equations. - Abstract: Using techniques from local bifurcation theory, we prove the existence of various types of temporally periodic and quasi-periodic waves for damped wave and beam equations, in higher dimensions. The emphasis is on understanding the role of external bifurcation parameters and symmetry, in generating the periodic/quasi-periodic motion of interest. Most of the work presented is joint with Brian

Pigott.

Date: **Friday, September 22, 2017.**

- Speaker:
**Gieri Simonett, Vanderbilt University.** - Title: Maximal regularity and critical spaces for quasilinear parabolic evolution equations.
- Abstract: In this talk, I will review some recent results concerning the topics listed in the title.

Applications to problems in fluid dynamics will be mentioned.

Date: **Friday, October 6, 2017.**

- Speaker:
**Glenn Webb, Vanderbilt University.** - Title: Spatial Spread of Epidemic Diseases in Geographical Settings: Seasonal Influenza Epidemics in Puerto Rico.
- Abstract: Deterministic models are developed for the spatial spread of epidemic diseases in geographical settings. The models are focused on outbreak that arise from a small number of infected hosts imported into sub-regions of the geographical settings. The goal is to understand how spatial heterogeneity influences the transmission dynamics of the susceptible and infected populations. The models consist of systems of partial differential equations with diffusion terms describing the spatial spread of the underlying microbial infectious agents. Applications are given to seasonal influenza epidemics in Puerto Rico.

Date: **Friday, Oct 20, 2017 (meeting at SC 1206).**

- Speaker:
**Yixiang Wu, Vanderbilt University.** - Title: On a reaction-diffusion vector-host epidemic model.
- Abstract: In this talk, I will present a reaction-diffusion vector-host epidemic model. We define the basic reproduction number R_0 and show that R_0 is a threshold parameter: if R_0 < 1 the disease free steady state is globally stable; if R_0 > 1 the model has a unique globally stable positive steady state. We then write R_0 as the spectral radius of the product of one multiplicative operator R(x) and two compact operators with spectral radius equalling one. Here R(x) corresponds to the basic reproduction number of the model without diffusion and is thus called the local basic reproduction number. We study the relationship between R_0 and R(x) as the diffusion rate varies.

Date: **Friday, Oct 27, 2017.**

- Speaker:
**Zehua Zhao, Johns Hopkins University.** - Title: Global well-posedness and scattering for the defocusing cubic Schrödinger equation on waveguide ℝ
^{2}x 𝕋^{2}. - Abstract: In this talk, we prove the large data scattering for the defocusing cubic nonlinear Schrödinger equation on waveguide ℝ
^{2}x 𝕋^{2}based on the assumption of resonant system conjecture, i.e. the large data scattering for the 2d cubic resonant system. More precisely, the “scattering threshold” for the cubic NLS and the cubic resonant system are same. This equation is critical both at the level of energy and mass. The key points are global-in-time Stricharz estimate, resonant system and profile decomposition.

Date: **Friday, Dec 1, 2017.**

- Speaker:
**Rares Rasdeaconu, Vanderbilt University.** - Title: Complex Manifolds and Special Hermitian Metrics.
- Abstract: In 2010, Streets and Tian introduced and studied a parabolic flow of pluriclosed metrics. Motivated by a still unsolved problem they raised, I will discuss several classes of hermitian metrics on closed complex manifolds and the relations between them. In particular, the equality between the balanced and the Gauduchon cones of metrics will be addressed. Time-permitting, I will describe the role played by the complex Monge-Ampere equation in the results presented. (Joint work with I. Chiose and I. Suvaina.)

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