We discuss concepts and review results about the Cauchy problem for the Fornberg-Whitham equation, which has also been called Burgers-Poisson equation in the literature. Our focus is on a comparison of various strong and weak solution concepts as well as on blow-up of strong solutions in the form of wave breaking. Along the way we add aspects regarding semiboundedness at blow-up, from semigroups of nonlinear operators to the Cauchy problem, and about continuous traveling waves as weak solutions.

中文翻译：

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Fornberg-Whitham 方程的解概念、适定性和破波

我们讨论有关 Fornberg-Whitham 方程（在文献中也称为 Burgers-Poisson 方程）的柯西问题的概念和结果。我们的重点是比较各种强解和弱解的概念，以及以破浪的形式爆发强解。在此过程中，我们添加了有关爆炸时半有界性的方面，从非线性算子的半群到柯西问题，以及关于作为弱解的连续行波。