## $$L^ 1$$ nonlinear stability of traveling waves for a hyperbolic system with relaxation.(English)Zbl 0879.35099

We investigate nonlinear stability of traveling wave solutions to the following (semilinear) hyperbolic $$2\times 2$$ system with a relaxation source term ${\partial u\over\partial t}+{\partial v\over\partial x}=0,\quad{\partial v\over\partial t}+\alpha {\partial u\over\partial x}= {1\over\varepsilon} (f(u)- v)\qquad (\varepsilon>0),$ where $$(x,t)\in\mathbb{R}\times \mathbb{R}_+$$. The unknowns $$u$$, $$v$$ belong to $$\mathbb{R}$$, the function $$f=f(u)$$ is in $$C^1(\mathbb{R})$$, and $$\alpha>0$$ is a fixed constant.

### MSC:

 35L60 First-order nonlinear hyperbolic equations 35B35 Stability in context of PDEs

### Keywords:

relaxation source term
Full Text: