The stability of travelling fronts for general scalar viscous balance law
The authors deal with the travelling front solutions of the viscous balance law \[ u_t+ (f(u))_x=\varepsilon u_{xx}+ g(u),\quad x\in\mathbb{R},\tag{1} \] where \(\varepsilon> 0\) is called the viscosity parameter. They are mainly interested in the existence and stability of different types of travelling waves for (1). Using shooting methods, the authors prove that there exist smooth monotone waves of (1) connecting adjacent zeroes of \(g\). Moreover, based on the existence results, by semigroup theory and detailed spectral analysis, the authors show that for every fixed \(\varepsilon> 0\) the travelling fronts for (1) are locally asymptotically stable in some suitably choosen weighted space, and especially for sufficiently small viscosity, one can choose the same weighted spaces in which the travelling fronts are uniformly exponentially stable.
scientific article; zbMATH DE number 2162032