Critical thresholds in a convolution model for nonlinear conservation laws (Q2782962)

The authors consider the following convolution model for the scalar conservation law NEWLINE\[NEWLINE u_t+uu_x=Q*u-u, NEWLINE\]NEWLINE where \(Q\) is a regular symmetric kernel, subject to initial data NEWLINE\[NEWLINE u(0,x)=u_0(x)\in C_b^1({\mathbb R}). NEWLINE\]NEWLINE The critical threshold phenomenon is shown by presenting a lower threshold for the breakdown of the solution and an upper threshold for the global existence of the smooth solution. These thresholds depend only on the relative size of the minimum slope of the initial velocity \(\inf\partial_x u_0\) and its maximal variation \(\sup u_0(x)-\inf u_0(x)\). The authors also establish the exact blow-up rate when the slope of the initial velocity is below the lower threshold. In the case when this slope is above of the upper threshold they prove the \(L^1\) time asymptotic stability of the smooth shock profile.