In this paper we study a $ 2\times2 $ semilinear hyperbolic system of partial differential equations, which is related to a semilinear wave equation with nonlinear, time-dependent damping in one space dimension. For this problem, we prove a well-posedness result in $ L^\infty $ in the space-time domain $ (0,1)\times [0,+\infty) $. Then we address the problem of the time-asymptotic stability of the zero solution and show that, under appropriate conditions, the solution decays to zero at an exponential rate in the space $ L^{\infty} $. The proofs are based on the analysis of the invariant domain of the unknowns, for which we show a contractive property. These results can yield a decay property in $ W^{1,\infty} $ for the corresponding solution to the semilinear wave equation.

中文翻译：

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关于有界域上一维半线性阻尼波动方程在 \begin{document}$ W^{1,\infty} $\end{document} 中的衰减

在本文中，我们研究了偏微分方程的 $2\times2 $ 半线性双曲系统，该系统与具有一维非线性时间相关阻尼的半线性波动方程相关。对于这个问题，我们在时空域 $ (0,1)\times [0,+\infty) $ 中证明了 $ L^\infty $ 的适定结果。然后我们解决零解的时间渐近稳定性问题，并表明在适当的条件下，解在空间 $ L^{\infty} $ 中以指数速率衰减到零。证明是基于对未知数不变域的分析，我们展示了它的收缩性质。对于半线性波动方程的相应解，这些结果可以产生以 $ W^{1,\infty} $ 表示的衰减特性。