The half-line boundary value problem for uxxx = ƒ(u)

Several existence and uniqueness results are obtained for third order problems of the form $\text{u}{}_{\mathrm{xxx}}\text{= ?(u), u(0)}\text{and}\text{u(∞)}$ specificied, plus one additional condition at x = 0. Using these results, we solve a nontrivial class of Riemann problems for a single conservation law, in the class of rarefraction waves plus discontinuities which are limits of fourth-order viscous profiles. For a nonconvex flux function, the solution so obtained differs, in many cases, from the classically “admissible” solution.