Autonomous and non-autonomous semilinear parabolic problems

自治和非自治半线性抛物线问题

• 批准号: EP/R023778/1
• 负责人: James Robinson
• 金额: $ 5.48万
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FR023778%2F1
• 关键词: Autonomous; non; autonomous; semilinear; parabolic

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.

偏微分方程是许多物理系统和过程的自然数学模型：热扩散，化学反应，流体流动和固体动力学。因此，它们在整个科学和工程中使用。然而，这些方程往往是复杂的，这是不寻常的，能够解决这些方程之一，“手”，以结束一个方便的公式。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”。考虑到这些方程在许多应用中的重要性，数学家自然会尽可能多地了解它们，即使（或者可能正是因为）我们在大多数情况下不能写下它们的解。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？".但这些问题可能很难回答，或者有一个答案，导致令人惊讶的新见解或观点。作为一个简单的例子，我们可以问方程x^2=-1是否有解。这里的答案取决于你想要变量x是什么;如果x必须是一个真实的数，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，人们可以引入一个新的概念（在复数的情况下，这个概念被证明是非常强大的）。另一个类似的问题是，一旦我们知道一个解决方案确实存在，这个解决方案是否是唯一的。同样，这个问题可能比它第一次出现时更微妙。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一个特定的解，这个方程确实有一个唯一的正解。类似的问题，尽管在一个更复杂的背景下，是偏微分方程数学理论的核心。给定一个模型，它是可能的，以表明确实有一个唯一的解决方案，然后可以继续调查进一步的性质，例如，是否解决方案是积极的，是否衰减到零，因为它改变了时间，是否“炸毁”在有限的时间.该项目旨在研究可以统一处理的广泛类别的模型;通过研究整体框架，而不是特定模型本身，可以证明一般结果，并查看它们如何从方程的底层结构中产生。然后，如果我们在我们的框架内分析各种模型方程，就有可能找出与这些方程相关的哪些现象来自它们的一般类，哪些更内在于我们正在研究的特定方程。这项工作将由沃里克的罗宾逊教授和罗德里格斯-伯纳尔教授合作完成，他将在休假期间从他在马德里的家乡学校来访。除了我们在这里的短期目标外，这次访问还将促进我们双方和我们的研究小组在今年结束后的长期合作。

项目成果

The heat flow in an optimal Fréchet space of unbounded initial data in R d

R d 中无界初始数据的最佳 Fréchet 空间中的热流

• DOI: 10.1016/j.jde.2020.07.017
• 发表时间: 2020
• 期刊: Journal of Differential Equations
• 影响因子: 2.4
• 作者: Robinson J

Linear Non-Autonomous Heat Flow in $$L_0^1({ {\mathbb {R}}}^{d})$$ and Applications to Elliptic Equations in $${ {\mathbb {R}}}^{d}$$

$$L_0^1({ {mathbb {R}}}^{d})$$中的线性非自治热流及其在$${ {mathbb {R}}}^{d}中椭圆方程的应用

• DOI: 10.1007/s10884-022-10195-6
• 发表时间: 2022
• 期刊: Journal of Dynamics and Differential Equations
• 影响因子: 1.3
• 作者: Robinson J

Estimates for the Heat Flow in Optimal Spaces of Unbounded Initial Data in $$\mathbb {R}^{ {d}}$$ and Applications to the Ornstein-Uhlenbeck Semigroup

$$mathbb {R}^{ {d}}$$ 中无界初始数据最优空间中的热流估计及其在 Ornstein-Uhlenbeck 半群中的应用

• DOI: 10.1007/s00009-023-02293-6
• 发表时间: 2023
• 期刊: Mediterranean Journal of Mathematics
• 影响因子: 1.1
• 作者: Robinson J

Optimal existence classes and nonlinear-like dynamics in the linear heat equation in R d

R d 线性热方程中的最优存在类和类非线性动力学

• DOI: 10.1016/j.aim.2018.06.009
• 发表时间: 2018
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Robinson J