A note on the Cauchy problem to a class of nonlinear dispersive equations with singular initial data
The Cauchy problem for the following class of nonlinear equations \[ \partial_t u +\partial_x f(u)+\delta C_n \partial_x^nu - \varepsilon \partial_x^2 u=0 \] is considered. The goal of the author is to generalize results by \textit{M. E. Schonbek} [Commun. Partial Differ. Equations 7, 959-1000 (1982; Zbl 0496.35058)] on the KdV-Burgers equation and by \textit{B. Cassis} [Commun. Partial Differ. Equations 14, 619-634 (1989; Zbl 0703.35043).] on a fourth-order equation. The unique, global-in-time, smooth solution is constructed for every initial condition from \(L^2({\mathbb R})\) and for the flux function \(f(u)\) of a quadratic growth. Moreover, for convex flux functions, the limit as \(\delta,\varepsilon\to 0\) of corresponding solutions is studied.
scientific article; zbMATH DE number 1491650