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\(L^ 1\) 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
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