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Stability of quasi-linear hyperbolic dissipative systems. (English) Zbl 0932.35024
This paper concerns $$2\pi$$-periodic in the space variables $$x_j$$ $$(j= 1,\dots, s)$$ solutions to \begin{aligned} \partial_tu &= \sum^s_{j=1} (A_{0j}+ \varepsilon A_{1j}(x,t,u,\varepsilon)) \partial_{x_j}u+ (B_0+\varepsilon B_1(x,t,u,\varepsilon))u \\ u(x,0) &= f(x)\end{aligned}\quad, x\in\mathbb{R}^n.\tag{1} Here $$A_{0j}$$ and $$B_j$$ are constant $$n\times n$$-matrices, and $$\varepsilon$$ is a small parameter.
It is assumed that there exists a $$\delta>0$$ such that for all $$\omega\in \mathbb{R}^n$$ the eigenvalues $$\lambda$$ of $i\sum^s_{j=1} A_{0j}\omega_j+ B_0$ satisfy $$\text{Re }\lambda\leq -\delta$$. Further, it is supposed that either the matrices $$A_{0j}$$, $$B_0$$ and $$A_{1j}$$ are Hermitian or that the eigenvalues of $i\sum^s_{j=1} (A_{0j}+ \varepsilon A_{1j}(x, t,u,\varepsilon))\omega_j$ are semi-simple and that their multiplicities do not depend on $$x$$, $$t$$, $$u$$, $$\varepsilon$$ and $$\omega$$.
The main result asserts that for any $$f$$ there exists an $$\varepsilon_0>0$$ such that for all $$\varepsilon\in [0,\varepsilon_0]$$ the solution to (1) converges uniformly to zero for $$t\to\infty$$.
Reviewer: L.Recke (Berlin)

##### MSC:
 35B35 Stability in context of PDEs 35B10 Periodic solutions to PDEs 35L60 First-order nonlinear hyperbolic equations 35L50 Initial-boundary value problems for first-order hyperbolic systems
##### Keywords:
stability of homogeneous stationary states
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##### References:
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