一类带不连续值的二维趋化系统解的全局存在性

Global Existence of a Two-Dimension Chemotaxis System with Discontinuous Data

摘要: 本文研究了描述肿瘤血管生成的二维趋化系统解的适定性和大时间行为。证明了当时间趋于无穷时，该系统的解收敛到一个常平衡态。同以往的结果相比，我们研究了不连续初值解的适定性。利用有效粘性通量和加权能量估计，得到了解的一系列先验估计，通过这些先验估计可以得到弱解的全局存在性。以往的研究都是关于连续初值的研究，得到的都是经典解。而实际情况中不连续的情形更加普遍，而本文的结果表明，在不连续初始值的情况下，我们能够得到二维趋化系统的全局弱解。

Abstract: This paper is concerned with the well-posedness and large-time behavior of a two-dimensional PDEODE hybrid chemotaxis system describing the initiation of tumor angiogenesis. We prove the solution converge to the constant equilibrium when the time tends to infinity. In contrast to the ex-isting related results, where continuous initial data is imposed, we are able to prove the asymptotic stability for discontinuous initial data with large oscillations. The key ingredient in our proof is the use of the so-called “effective viscous flux”, which enables us to obtain the desired energy estimates and regularity. The previous results are all about the continuous initial data and the classical solutions are obtained. However, discontinuity is more common in practice, and the results of this paper show that we can obtain the global weak solution of the two-dimensional chemotaxis system under the discontinuous initial data.