Existence and uniqueness of solutions of reaction-convection equations with non-Lipschitz nonlinearity

In this work we study single balance law ut+∇⋅Φ(u)=f(u) with bounded initial value, and find that there may exist maximal and minimal solutions, if f(u) is not Lipschitz continuous at u=0. We also show that comparison principle is valid for such solutions, and the solutions may blow up or not under certain conditions. It is determined by the strength of source supply, as well as the competition between the source and flux.