Asymptotic profile of solutions of conservation laws with source (Q1909695)

The author investigates the Cauchy problem for the following balance law: \(u_t+ f(u)_x= g(u)\), \(x\in \mathbb{R}\), \(t> 0\), with convex flux \(f\) and initial value with compact support. The source function \(g\in C^1\) is supposed to have a finite number of zeros with \(g(0)= 0\) and to satisfy \(g(v)v\leq \alpha v^2+ \beta\) for some \(\alpha\), \(\beta> 0\). It is proved, that the asymptotic profile of the generalized solution consists in a number of rarefaction waves divided by regions where the solutions oscillate around an unstable zero of the source term.