一类带分数阶扩散项的聚合方程的研究

In this project，we mainly study a class of aggregation equation with fractional dissipation, this equations arise in many problem in biology. It describes the competition between the tendency for organisms to diffuse and for organisms to aggregate into groups through nonlocal self-attraction. How the singularities of the kernel function influence the existence and nonexistence of global-in-time solution are studied. We also want to examine the long-time behavior of dissipation solutions and give the construction of self-similar solutions for certain kernel function. The fractional Laplace operator has an integral representation, which is difference with the differential operator. Thus give us a new difficulty. This project is motivated by such challenge. We would like to obtain analogous results for the aggregation equation with fractional dissipation. All the proofs are based on delicate estimates for the energy function. This study will contribute to our understanding of the aggregation behavior in nature, and also have great theoretical significance.

本项目拟对一类重要的带分数阶扩散项的聚合方程解的性质进行研究，这类方程有着深刻的生物学背景，它刻画了物种在扩散和聚合之间的竞争。研究内容主要包括：方程中核函数梯度的奇异性对解的局部存在，整体存在和有限时间爆破的影响；对于特殊的核函数，研究解的长时间行为和自相似解。由于分数阶扩散项是采用积分形式来表示，这不同于传统的二阶微分算子，这给研究带来了新的挑战，这也是促使我们研究的动力。本项目主要目的是将已有文献中的部分结论推广到含分数阶扩散项的情形。通过对已有文献中的方法技巧的总结，我们主要通过对能量泛函的更精细的先验估计来研究方程解的性质。对该类模型的研究不仅有助于我们认识自然界中出现的聚合现象，了解其形成和演化的过程，同时也有重大的理论意义。

深入研究带分数阶Laplace算子的发展方程的动力学行为是一个有着重要意义和丰富内容的课题，最近几年来对带分数阶Laplace算子的聚合方程吸引了许多数学工作者进行研究。所涉及的问题包括解的存在性，爆破条件，解的长时间动力学行为等。在本项目的执行过程中，对一类带分数阶Laplace算子的聚合方程的解的性质进行了研究，我们把Grzegorz Karch, Kanako Suzuki在论文《Blow-up versus global existence of solutions to aggregation equations》的结论推广到了带分数阶Laplace算子的情形。得到的结论有助于加强人们对相应模型的认识理解，并丰富偏微分方程的理论。因此，有着重要的理论意义和一定的学术价值。