In this paper, we consider the inverse problem of determining some coefficients within a coupled nonlinear parabolic system, through boundary observation of its non-negative solutions. In the physical setup, the non-negative solutions represent certain probability densities in different contexts. We innovate the successive linearisation method by further developing a high-order variation scheme which can both ensure the positivity of the solutions and effectively tackle the nonlinear inverse problem. This enables us to establish several novel unique identifiability results for the inverse problem in a rather general setup. For a theoretical perspective, our study addresses an important topic in partial differential equation (PDE) analysis on how to characterise the function spaces generated by the products of non-positive solutions of parabolic PDEs. As a typical and practically interesting application, we apply our general results to inverse problems for ecological population models, where the positive solutions signify the population densities.

中文翻译：

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通过生物学应用对其非负解的边界观察来确定抛物线系统

在本文中，我们考虑通过对其非负解的边界观察来确定耦合非线性抛物线系统中某些系数的反问题。在物理设置中，非负解代表不同上下文中的某些概率密度。我们创新了逐次线性化方法，进一步发展了高阶变分格式，既能保证解的正性，又能有效解决非线性逆问题。这使我们能够在相当通用的设置中为反问题建立几个新颖的独特可识别性结果。从理论角度来看，我们的研究解决了偏微分方程 (PDE) 分析中的一个重要主题，即如何表征由抛物型 PDE 的非正解乘积生成的函数空间。作为一个典型且实际有趣的应用，我们将一般结果应用于生态种群模型的反问题，其中正解表示种群密度。