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

  • image(1)

is studied. Various self-similar solutions exhibiting single point gradient blow-up in finite time, as t[RIGHTWARDS ARROW]T < ∞, 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)

  • image(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

  • image(3)

with the standard linear integral operator (−D2x)−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 u0(x)).