Autonomous and non-autonomous semilinear parabolic problems

偏微分方程是许多物理系统和过程的自然数学模型：热的扩散、化学反应、流体的流动和固体的动力学。因此，它们在整个科学和工程中都被使用。然而，这些方程通常是复杂的，并且能够“手工”计算出其中一个方程的解并最终得到一个方便的公式是不寻常的。在实际应用中，这些方程通常在计算机上近似，然后用这种方法得到近似的“解”。考虑到这些方程在如此多的应用中是如此重要，数学家们很自然地试图尽可能多地理解它们，即使（或者可能正是因为）我们在大多数情况下无法写下它们的解。数学家提出的许多问题乍一看似乎微不足道，比如“这个方程有解吗？”但这样的问题很难回答，或者有一个答案会导致令人惊讶的新见解或观点。作为一个简单的例子，人们可以问方程x^2=-1是否有解。这里的答案取决于你想让变量x是什么；如果x一定是实数，那么答案是否定的（因为任何平方都是正数）；但是如果你允许x是一个复数，答案是肯定的，通过首先提出这个问题，可以引入一个新概念（在复数的情况下，这个概念变得非常强大）。另一个类似层次的问题是，一旦我们知道一个解存在，这个解是否唯一。同样，这个问题可能比它最初出现时更加微妙。方程x^2=1有唯一解吗？不，因为x=-1和x=+1都是解。但是如果我们想要一个定义明确的方法来选择特解，这个方程确实有一个唯一的正解。类似的问题存在于偏微分方程数学理论的核心，尽管背景更为复杂。给定一个可以证明确实存在唯一解的模型，人们就可以继续研究进一步的性质，例如，解是否为正，是否随着时间的变化而衰减为零，是否在有限时间内“爆炸”……这个项目旨在研究一大类可以统一处理的模型；通过研究总体框架，而不是特定模型本身，可以证明一般结果，并了解它们是如何从方程的底层结构中产生的。然后，如果我们在我们的框架内分析各种模型方程，就有可能弄清楚哪些与这些方程相关的现象是由它们的一般类别产生的，哪些是我们正在研究的特定方程的内在特征。这项工作将由华威大学的罗宾逊教授和Rodriguez-Bernal教授合作完成，Rodriguez-Bernal教授将在马德里的家乡机构休假期间访问。除了我们的短期目标外，这次访问还将在今年结束后很长一段时间内促进我们双方和我们的研究小组之间的长期合作。

Partial differential equations are the natural mathematical models for many physical systems and processes: diffusion of heat, chemical reactions, the flow of fluids, and the dynamics of solids. As such they are used throughout the sciences, and in engineering. However, these equations are often complicated, and it is unusual to be able to work out the solution of one of these equations "by hand" to end up with a convenient formula. More usually in actual applications these equations are approximated on a computer, and then an approximate "solution" is obtained this way.Given how important these equations are in so many applications, it is natural for mathematicians to try to understand as much as they can about them, even if (or perhaps precisely because) we cannot in most cases write down their solution. Many of the questions mathematicians ask may even seem trivial at first sight, such as "does this equation have a solution?". But such questions can be hard to answer, or have an answer that leads to surprising new insights or points of view. As a simple illustration, one can ask whether the equation x^2=-1 has a solution. Here the answer depends on what you want the variable x to be; if x must be a real number then the answer is no (since any square is positive); but if you allow x to be a complex number the answer is yes, and by asking the question in the first place one can be lead to introduce a new concept (which in the case of complex numbers turned out to be incredibly powerful). Another question at a similar level is, once we know that a solution does exist, whether or not this solution is unique. Again, this question can be more subtle that it first appears. Does the equation x^2=1 have a unique solution? No, since x=-1 and x=+1 are both solutions. But if we want a well-defined way to choose a particular solution, this equation does have a unique positive solution.Similar questions, albeit in a more complicated setting, lie at the heart of the mathematical theory of partial differential equations. Given a model for which it is possible to show that there is indeed a unique solution, one can then go on to investigate further properties, e.g. whether the solution is positive, whether it decays to zero as it changes in times, whether it "blows up" in a finite time...This project aims to look at a wide class of models that can be treated in a unified way; by studying the over-arching framework, rather than the particular models themselves, it becomes possible to prove general results and see how they arise from the underlying structure of the equations. Then, if we analyse various model equations within our framework, it becomes possible to work out which phenomena associated with these equations arise from their general class and which are more intrinsic to the particular equation we are studying.The work will be done as a collaboration between Prof. Robinson, based at Warwick, and Prof. Rodriguez-Bernal, who will be visiting while on a sabbatical year from his home institution in Madrid. As well as our short-term goals here this visit will foster longer-term collaboration between us both and our research groups long after the year is over.