Global Existence in Reaction-Diffusion Systems with Control of Mass: a Survey
Global Existence in Reaction-Diffusion Systems with Control of Mass: a Survey
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DOI：
10.1007/s00032-010-0133-4
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发表时间：
2010-12-01
影响因子：
1.7
通讯作者：
Pierre, Michel
中科院分区：
文献类型：
--
作者：
Pierre, Michel
本文的目的是描述具有两个主要性质的反应扩散方程组解的整体存在性问题的研究现状：一方面，解的正性始终保持不变；另一方面，各分量的总质量在时间上是一致控制的。这种对质量的统一控制(或者--用数学术语说--关于解的L(1)范数)表明，在有限时间内不应该发生爆炸。事实证明，情况并不是那么简单。这就解释了为什么关于这一主题的文献中有这么多不同方向的部分结果，也解释了为什么在具有这两个自然性质的应用中出现了许多具有这两个自然性质的系统，但整体存在的一般问题仍然是开放的。在这里，我们回顾了关于全球存在的主要的积极和消极的结果，以及许多参考文献，对仍然悬而未决的问题的描述和一些新的结果。
The goal of this paper is to describe the state of the art on the question of global existence of solutions to reaction-diffusion systems for which two main properties hold: on one hand, the positivity of the solutions is preserved for all time; on the other hand, the total mass of the components is uniformly controlled in time. This uniform control on the mass (or - in mathematical terms-on the L(1)-norm of the solution) suggests that no blow up should occur in finite time. It turns out that the situation is not so simple. This explains why so many partial results in different directions are found in the literature on this topic, and why also the general question of global existence is still open, while lots of systems arise in applications with these two natural properties. We recall here the main positive and negative results on global existence, together with many references, a description of the still open problems and a few new results as well.