## Mathematics & Statistics ETDs

9-5-2013

#### Abstract

The dissertation consists of two parts, Well-posedness and ill-posedness for the nonlinear beam equation and Strichartz estimates of the beam equation on the domains. In the first part, we will work to introduce the further studies of Strichartz estimates with initial data both in homogeneous Sobolev spaces $\\dot{H}^s\ imes\\dot{H}^{s-2}$ and in inhomogeneous Sobolev space ${H}^s\ imes{H}^{s-2}$. We take advantage of the Strichartz estimates to build well-posedness theorems of the nonlinear beam equations for rough data by the Picard iteration method. We will apply these methods on the nonlinear beam equation with energy critical, subcritical" and energy supercritical" cases. Since the beam equation does not satisfy finite speed propagation, we introduce the further result of the fractional chain rule to deal with the energy super critical" case. We obtain the global well-posedness with initial data in homogeneous Sobolev space $\\dot{H}^s\ imes\\dot{H}^{s-2}$ and local well-posedness with initial data in inhomogeneous Sobolev space ${H}^s\ imes{H}^{s-2}$. At the same time, we extend the range of order $s$. With the global existence for small data, we prove the scattering and asymptotic completeness result for the nonlinear beam equation. Last we prove the nonlinear beam equation is ill-posed in defocusing case $\\omega=-1$ when \$ 0

Mathematics

Doctoral

#### Department Name

Mathematics & Statistics

Blair, Matthew

Blair, Matthew

#### Second Committee Member

Pereyra, M. Cristina

Lorenz, Jens

#### Fourth Committee Member

Metcalfe, Jason L

English

#### Keywords

Strichartz estimates, Well-posedness, Ill-posedness, beam equation

Dissertation

COinS