Single point gradient blow-up and nonuniqueness for a third-order nonlinear dispersion equation

A basic mechanism of a formation of shocks via gradient blow-up from analytic solutions for the third-order nonlinear dispersion PDE from compacton theory u(t) = (uu(x))(xx) in R x R+, (1) is studied. Various self-similar solutions exhibiting single point gradient blow-up in finite time, as t -> T- < infinity, with locally bounded final time profiles u(x, T-), are constructed. These are shown to admit infinitely many discontinuous shock-type similarity extensions for t > T, all of them satisfying generalized Rankine-Hugoniot's condition at shocks. As a consequence, the nonuniqueness of solutions of the Cauchy problem after blow-up is detected. This is in striking difference with general uniqueness-entropy theory for the 1D conservation laws such as (a partial differential equation, PDE, Euler's equation from gas dynamics) u(t) + uu(x) = 0 in R x R+, (2) created by Oleinik in the middle of the 1950s. Contrary to (1) and not surprisingly, self-similar gradient blow-up for (2) is shown to admit a unique continuation. Bearing in mind the classic form (2), the NDE (1) can be written as (-D-x(2))(-1) u(t) + uu(x) = 0 (aconservation law in H-1(R)), (3) with the standard linear integral operator (-D-x(2))(-1) > 0. However, because (3) is a nonlocal equation, no standard entropy and/or BV-approaches apply (moreover, the x-variations of solutions of (3) is increasing for BV data u(0)(x)).