AI 中文总结
该研究针对扩散-弛豫系统，通过能量耗散结构、不变区域估计及几何奇异摄动理论，严格刻画了平衡化与局域化的参数转变条件及对应解的行为。
AI 中文摘要
我们研究一类扩散-弛豫系统，探讨导致平衡化与局域化的参数条件：当扩散占主导时，解收敛到均匀平衡态；反之，当有效扩散较弱时，会出现局域化现象。这类行为已通过形式渐近分析和线性化稳定性分析在多种模型中得到研究，但对相关非线性现象的严格理解仍有限，尤其在高维情形下。在平衡化区域，我们利用能量耗散结构和不变区域估计，建立解向常数平衡态的收敛性；在局域化区域，我们研究自相似解，将其存在性问题转化为自治动力系统问题，与局域化解关联的自相似剖面的存在性，归结为构造自治动力系统的异宿轨道，其存在性通过几何奇异摄动理论的应用得到证明。我们的分析对从平衡化到局域化的转变提供了严格刻画。
英文摘要
We consider a diffusion-relaxation system and investigate the conditions on parameters leading to equilibration versus localization. When the diffusion is dominant, solutions converge toward homogeneous equilibria. By contrast, when the effective diffusion is weak, localization emerges. Such behaviors have been studied for various models through formal asymptotic arguments and linearized stability analysis, but rigorous understanding of the associated nonlinear phenomena remains limited, particularly in higher dimensions. In the equilibration regime, we establish convergence toward constant equilibria by exploiting an energy dissipation structure and invariant-region estimates. In the localization regime, we study self-similar solutions and transform the problem of their existence into an autonomous dynamical system. The existence of self-similar profiles associated to localizing solutions is reduced to the construction of a heteroclinic orbit for an autonomous dynamical system. Their existence is obtained through an application of geometric singular perturbation theory. Our analysis provides a rigorous characterization of the transition from equilibration to localization.
