具有非线性扩散的反应扩散方程组的整体重整化解与平衡
Global renormalised solutions and equilibration of reaction-diffusion systems with non-linear diffusion
摘要：
研究了一类具有非线性扩散的反应扩散方程组重正解的整体存在性和收敛于平衡点的条件。假设系统具有拟正非线性，并满足熵不等式。通过引入一类新的加权截断函数，克服了由退化扩散引起的全局重正化解的困难。利用所得到的整体重整化解，我们研究了具有非线性扩散的化学反应网络理论中复杂平衡系统的大时间行为。当反应网络不存在边界平衡点时，利用熵方法证明了复平衡点指数吸引同一相容类中的所有重整化解.这种收敛甚至延伸到一个范围内的非线性扩散，在全球存在是一个开放的问题，但我们能够证明，近似系统的解决方案，以指数收敛到平衡均匀的正则化参数。
英文摘要：
The global existence of renormalised solutions and convergence to equilibrium for reaction-diffusion systems with non-linear diffusion are investigated. The system is assumed to have quasi-positive non-linearities and to satisfy an entropy inequality. The difficulties in establishing global renormalised solutions caused by possibly degenerate diffusion are overcome by introducing a new class of weighted truncation functions. By means of the obtained global renormalised solutions, we study the large-time behaviour of complex balanced systems arising from chemical reaction network theory with non-linear diffusion. When the reaction network does not admit boundary equilibria, the complex balanced equilibrium is shown, by using the entropy method, to exponentially attract all renormalised solutions in the same compatibility class. This convergence extends even to a range of non-linear diffusion, where global existence is an open problem, yet we are able to show that solutions to approximate systems converge exponentially to equilibrium uniformly in the regularisation parameter.
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