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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
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