Global existence vs shock formation in the one-dimensional fractal Burgers equation

We study properties of solutions of the initial value problem for the nonlinear and nonlocal equation $u_t+(-\partial ^2_x)^{\alpha/2} u+uu_x=0$ with $\alpha \in (0,1)$, supplemented with an initial datum approaching the constant states $u_\pm $ ($u_- \gt u_+$) as $x\to \pm \infty $, respectively. For “small” initial data, we prove regularity of solutions and their asymptotics. For “large” initial data we prove the blow-up of $u_x$ in finite time. Our results are comparable to those of [N. Alibaud et al., J. Hyperbolic Differential Equations 4 (2007), 479–499]; however, we prove them by different methods.