Klein–Saut conjecture on blow-up and global existence for dispersive Burgers equations

Let $u$ solve the dispersive Burgers-type equation considered above, with dispersion parameter $\alpha$, and let $T^*$ denote a finite blow-up time. A solution undergoes wave breaking when $\|u(t)\|_{L^\infty}$ remains bounded while $\|\partial_xu(t)\|_{L^\infty}\to+\infty$.

Klein–Saut conjecture. The following assertions hold:

1. If $\alpha\leq 1$, solutions blow up in finite time through wave breaking:

$$\lim_{t\to T^*}\|u(t)\|_{L^\infty}\text{ stays bounded},\qquad \|\partial_xu(t)\|_{L^\infty}\to+\infty.$$

2. If $\alpha>1$, sufficiently small initial data give rise to global-in-time solutions.
3. If $1<\alpha\leq\frac{3}{2}$, large solutions blow up in finite time through a dispersive blow-up scenario, with

$$\|u(t)\|_{L^\infty_x}\to+\infty.$$

4. If $\alpha>\frac{3}{2}$, solutions exist globally in time.

These predictions concern the low-dispersion regime and were motivated by numerical work. The full set of assertions is presented as a conjectural description of the transition between wave breaking, dispersive blow-up, and global existence.