New Trends in Quantum and Classical Kinetic Equations and Related PDEs

Higher order nonlinear dispersive equation on a quarter plane

Hongqiu Chen

University of Memphis

The focus of the talk is the higher order nonlinear dispersive equation \begin{equation*}\begin{split}%\label{5thbbm} u_t+u_x-\gamma_1\beta{u}_{xxt}+\gamma_2{u}_{xxx}+\delta_1\beta^2{u}_{xxxxt}+\delta_2u_{xxxxx}+\frac34\alpha(u^2)_x \\ + \alpha\beta\Big(\gamma (u^2)_{xx}-\frac{7}{48}u_x^2\Big)_x-\frac18\alpha^2(u^3)_x=0 \end{split}\end{equation*} which models unidirectional propagation of small amplitude long waves in dispersive media. The dependent variable $u=u(x, t)$ is a real-valued function of $x, t.$ %\in \mathbb{R} \,\, {\rm and} \,\, t\geq 0.$ It represents the deviation of the free surface relative to its undisturbed state at the space point $x$ and at time $t.$ The subscripts connote partial derivatives while $\gamma_1, \delta_1, \alpha,\beta>0$ and $ \gamma_2, \delta_2, \gamma \in\mathbb R$ are modeling constants. The specific interest of this talk is in the initial-boundary value problem where both spatial and time variables lie in $\mathbb R^+,$ namely, quarter plane problem. With proper requirements on initial and boundary condition, we show local and global well posedness.