In this article, we introduce the notion of relative $\mathcal {K}$-equi-stability (RKES) to characterize the uniformly continuous dependence of (weak) solutions on external disturbances for nonlinear parabolic partial differential equations (PDEs). Based on the RKES, we prove the input-to-state stability (ISS) in the spatial sup-norm for a class of nonlinear parabolic PDEs with either Dirichlet or Robin boundary disturbances. An example concerned with a superlinear parabolic PDE with Robin boundary condition is provided to illustrate the obtained ISS results. Besides, as an application of the notion of RKES, we conduct stability analysis for a class of parabolic PDEs in cascade coupled over the domain or on the boundary of the domain, in the spatial and time sup-norm, and in the spatial sup-norm, respectively. The technique of De Giorgi iteration is extensively used in the proof of the results presented in this article.

中文翻译：

---

抛物线 PDE 的 Sup-Norm 的相对稳定性和空间 Sup-Norm 的输入到状态稳定性

在这篇文章中，我们介绍了相对的$\数学{K}$- 等稳定性 (RKES) 来表征非线性抛物型偏微分方程 (PDE) 的（弱）解对外部扰动的一致连续依赖性。基于 RKES，我们证明了具有 Dirichlet 或 Robin 边界扰动的一类非线性抛物线偏微分方程在空间上范数中的输入到状态稳定性 (ISS)。提供了一个有关具有 Robin 边界条件的超线性抛物线偏微分方程的示例来说明获得的 ISS 结果。此外，作为 RKES 概念的应用，我们对域上或域边界上级联耦合的一类抛物线偏微分方程，在空间和时间上范数，以及在空间支持上进行稳定性分析。规范，分别。