Third-order nonlinear dispersive equations: Shocks, rarefaction, and blowup waves

Shock waves and blowup arising in third-order nonlinear dispersive equations are studied. The underlying model is the equation in
\begin{equation} u_t=(uu_x)_{xx}\quad\text{in}\quad\mathbb R\times\mathbb R_+. \label{1} \end{equation}
It is shown that two basic Riemann problems for Eq. (1) with the initial data
$$ S_{\pm}(x)=\mp\operatorname{sign}{x} $$
exhibit a shock wave ($u(x,t)\equiv S_{-}(x)$) and a smooth rarefaction wave (for $S_{+}$), respectively. Various blowing-up and global similarity solutions to Eq. (0.1) are constructed that demonstrate the fine structure of shock and rarefaction waves. A technique based on eigenfunctions and the nonlinear capacity is developed to prove the blowup of solutions. The analysis of Eq. (1) resembles the entropy theory of scalar conservation laws of the form $u_t+uu_x=0$, which was developed by O. A. Oleinik and S. N. Kruzhkov (for equations in $x\in\mathbb R^N$ ) in the 1950s–1960s.