Critical thresholds in a convolution model for nonlinear conservation laws

In this work we consider a convolution model for nonlinear conservation laws. Due to the delicate balance between the nonlinear convection and the nonlocal forcing, this model allows for narrower shock layers than those in the viscous Burgers equation and yet exhibits the conditional finite time breakdown as in the damped Burgers equation. We show the critical threshold phenomenon by presenting a lower threshold for the breakdown of the solutions and an upper threshold for the global existence of the smooth solution. The threshold condition depends only on the relative size of the minimum slope of the initial velocity and its maximal variation. We show the exact blow-up rate when the slope of the initial pro le is below the lower threshold. We further prove the L 1 stability of the smooth shock pro le, provided the slope of the initial pro le is above the critical threshold.