Abstract We study an ill-posed forward-backward parabolic equation with techniques of nonstandard analysis. Equations of this form arise in many applications, ranging from the description of phase transitions, to the dynamics of aggregating populations, and to image enhancing. The equation is ill-posed in the sense that it only has measure-valued solutions that in general are not unique. By using grid functions of nonstandard analysis, we derive a continuous-in-time and discrete-in-space formulation for the ill-posed problem. This nonstandard formulation is well-posed and formally equivalent to the classical PDE, and has a unique grid solution that satisfies the properties of an entropy measure-valued solution for the original problem. By exploiting the strength of the nonstandard formulation, we are able to characterize the asymptotic behaviour of the grid solutions and to prove that they satisfy a conjecture formulated by Smarrazzo for the measure-valued solutions to the ill-posed problem.

中文翻译：

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一类不适定抛物线方程的网格函数公式

摘要 我们用非标准分析技术研究了一个不适定的前向后向抛物线方程。这种形式的方程出现在许多应用中，从相变的描述到聚集种群的动力学，再到图像增强。该方程是不适定的，因为它只有通常不是唯一的测量值解。通过使用非标准分析的网格函数，我们为不适定问题推导出时间连续和空间离散的公式。这种非标准公式是适定的，形式上等同于经典的 PDE，并且具有唯一的网格解，该解满足原始问题的熵测度值解的性质。通过利用非标准配方的优势，