Eventually positive operator semigroups and their application to evolution equations

许多动力系统的时间演化具有被称为“正性”或“正性保持”的性质，其含义如下：如果系统的初始值（例如，它可以是一个具有多个分量的向量，或者是一个定义在某个集合上的函数）在每个分量或每个点上都是非负的，则该特性对于系统在每个后续时间的轨迹也是成立的。这种行为在许多物理、化学或概率模型中自然发生（例如，质量分布或概率分布本质上是非负的）。如果这样的动力系统也是线性和自治的，从数学的观点来看，它可以用所谓的“正算子半群”来描述。到目前为止，这种正半群已经有了一个深刻而全面的理论。最近，在一些偏微分方程中观察到了更复杂的“最终正性”现象。这种现象意味着，对于非负的初始值，系统的轨迹可以在很小的时间内改变符号，但它再次成为并保持非负的所有足够大的时间。在有限维度的早期结果之后，最终积极性的一般无限维理论在2016年的两篇文章中提出，并从那时起由多位作者开发和应用。因此，现在已知的微分方程有很多种最终的正解。尽管如此，该理论的艺术水平还远远不够全面。它留下了许多悬而未决的问题--其中一些在正半群的情况下很久以前就已经回答了--对于许多具体的微分方程来说，仍然不可能通过目前可用的理论来确定它们的解最终是否为正。该项目旨在开发新的方法，以便在理论层面上更好地理解最终正性，并在进一步的具体微分方程中识别最终正性。我们将使用来自泛函分析各个领域的工具，特别是强连续算子半群和Banach格，来证明最终正性的新特征和新的充分条件，我们将把这些结果应用于各种微分方程，以确定它们的解是否最终是正的。

The time evolution of many dynamical systems has a property which is referred to as "positivity" or "preservation of positivity" and which means the following: if the initial value of the system (which can, for instance be a vector with finitely many components, or a function defined on some set) is non-negative in each component or at each point, then this property is also true for the trajectory of the system at each subsequent time. This behaviour occurs naturally in many physical, chemical or probabilistic models (as, for instance, mass distributions or probability distributions are inherently non-negative). In case that such a dynamical system is also linear and autonomous, it can, from a mathematical point of view, be described by a so-called "positive operator semigroup". As of today, a deep and thorough theory of such positive semigroups is available. More recently, the more involved phenomenon of "eventual positivity" has been observed in a number of partial differential equations. This phenomenon means that, for non-negative initial value, the trajectory of the system can change sign for small times, but it again becomes and stays non-negative for all sufficiently large times. Following earlier results in finite dimensions, a general infinite-dimensional theory of eventual positivity was initiated in two articles in 2016 and was from then on developed and applied by various authors. Thus, a large variety of differential equations with eventually positive solutions is nowadays known. Still, the state of the art of the theory is far from comprehensive. It leaves many questions open - some of which have been answered long ago in the case of positive semigroups -, and for many concrete differential equations it is still not possible by means of the currently available theory to determine whether their solutions are eventually positive. This project aims at developing new methods in order to better understand eventual positivity at a theoretical level and to identify eventual positivity in further concrete differential equations. We will employ tools from various fields in functional analysis, in particular strongly continuous operator semigroups and Banach lattices, to prove new characterizations and new sufficient conditions for eventual positivity, and we will apply these results to a variety of differential equations to determine whether their solutions are eventually positive.