This paper is concerned with global existence of classical solutions as well as occurrence of infinite-time blowups to the following fully parabolic system

in a smooth bounded domain \(\varOmega \subset {\mathbb {R}}^n\), \(n\ge 1\) with no-flux boundary conditions. This model was recently proposed in Fu et al. (Phys Rev Lett 108:198102, 2012) and Liu et al. (Science 334:238, 2011) to describe the process of stripe pattern formations via the so-called self-trapping mechanism. The system features a signal-dependent motility function \(\gamma (\cdot )\), which is decreasing in v and will vanish as v tends to infinity. An essential difficulty in analysis comes from the possible degeneracy as \(v\nearrow \infty .\) In this work we develop a novel comparison method to tackle the degeneracy issue, which greatly differs from the conventional energy method in literature. An explicit point-wise upper-bound estimate for v is obtained for the first time, which shows that v(x, t) grows point-wisely at most exponentially in time. An intrinsic mechanism is then unveiled that the finite-time degeneracy is prohibited in any spatial dimension with a generic decreasing \(\gamma \). With such new findings, we further study global existence of classical solutions when \(n\le 3\) and discuss uniform-in-time boundedness when \(\gamma (\cdot )\) decreases algebraically at large signal concentrations. Besides, a new critical-mass phenomenon in dimension two is observed if \(\gamma (v)=e^{-v}\). Indeed, we prove that the classical solution always exists globally and remains uniformly-in-time bounded in the sub-critical case, while in the super-critical case a blowup may take place in infinite time rather than finite time.

本文关注经典解的全局存在以及以下完全抛物线系统的无限时间爆破的发生

在具有无通量边界条件的光滑有界域\（\ varOmega \ subset {\ mathbb {R}} ^ n \），\（n \ ge 1 \）中。Fu等人最近提出了这种模型。（Phys Rev Lett 108：198102，2012）和Liu等人。（Science 334：238，2011）来描述通过所谓的自陷机制形成条纹图案的过程。该系统具有与信号相关的运动函数\（\ gamma（\ cdot）\），该函数在v中减小，并且随着v趋于无穷大而消失。分析中的一个基本困难来自可能的简并为\（v \ nearrow \ infty。\）在这项工作中，我们开发了一种新颖的比较方法来解决简并性问题，这与文献中的常规能量方法大不相同。首次获得v的明确的逐点上限估计，这表明v（x，  t）在时间上最多逐点增长。然后揭示了一种内在机制，即在任何空间维度上都禁止以通用递减\（\ gamma \）进行有限时间退化。有了这样的新发现，我们将进一步研究\（n \ le 3 \）时经典解的整体存在性，并讨论\（\ gamma（\ cdot）\）时的时间均匀有界性。在大信号集中，代数递减。此外，如果\（\ gamma（v）= e ^ {-v} \），则会在第二维上观察到新的临界质量现象。实际上，我们证明了经典解决方案始终存在于全局范围内，并且在次临界情况下始终保持时间一致的边界，而在超临界情况下，爆炸可能会在无限时间内而不是有限时间内发生。