Blow-up phenomena for a family of Burgers-like equations

By introducing a stress multiplier we derive a family of Burgers-like equations. We investigate the blow-up phenomena of the equations both on the real line R and on the circle S to get a comparison with the Degasperis–Procesi equation. On the line R, we first establish the local well-posedness and the blow-up scenario. Then we use conservation laws of the equations to get the estimate for the L∞-norm of the strong solutions, by which we prove that the solutions to the equations may blow up in the form of wave breaking for certain initial profiles. Analogous results are provided in the periodic case. Especially, we find differences between the Burgers-like equations and the Degasperis–Procesi equation, see Remark 4.1.