Effects of Anomalous Diffusion on Pattern Formation and Nonlinear Dynamics in Reaction-Diffusion systems, and Applications

主要研究人员和他们的同事开发了异常扩散系统中的模式形成、非线性动力学和传输理论，并将其应用于许多重要问题，例如模式形成反应扩散问题和药物输送问题。与规则扩散不同，在规则扩散中，随机行走粒子的均方位移对时间的依赖性是线性的，在异常扩散中，均方位移表现为时间的幂函数。如果指数小于1，则扩散过程比正常扩散慢，称为亚扩散，而如果指数超过1，则比正常扩散快，称为超扩散。 反常扩散的数学描述涉及积分微分算子，其必须从适当的连续时间随机游走模型导出，并且难以研究。具体而言，研究人员研究（i）增长域中的模式形成，包括正常扩散和异常扩散，重点是当某些扩散系数渐近小时的奇异扰动情况;（ii）反应异常扩散系统中的图灵模式选择，特别是条纹或斑点的选择;（iii）反应-超扩散系统中的局域态，其出现是由于例如条纹和时间周期单元之间的前沿钉扎现象;和（iv）药物输送问题，其中包括一个近似的分析理论的发展subdiffusive问题与移动的自由边界;对受化学品亚扩散运输控制的生物可侵蚀的受控药物递送装置的模型的研究，在电场存在下的透皮药物释放，即伴随着离子电渗疗法等。多年来，药物控制递送作为一种有效的治疗多种疾病的方法在医学界引起了极大的关注。各种药物输送装置所基于的共同原理是给定药物向特定器官的质量传递，其中质量传递速率或位置或两者根据某些医疗协议规定。在设计和开发各种受控药物递送系统方面已经取得了很大进展，许多人常规服用为受控释放设计的药物。药物传递系统的数学建模是非常重要的，因为它可以提供更好的理解和定量描述的物理，化学和生物过程控制的系统的性能。在此描述的基础上，可以设计更好的受控药物递送系统。有实验证据表明，药物向生物靶标的扩散不是正常的，而是非常缓慢的，所谓的亚扩散，因为药物分子必须通过非常拥挤的环境扩散。研究人员研究受亚扩散控制的药物转运，以便更好地了解药物递送过程。此外，研究人员还研究了其他类型的问题，其中亚扩散和超扩散（比正常扩散快）很重要。这是反应扩散系统中的模式形成。亚扩散经常发生在多孔介质和聚合物中，而超扩散则是等离子体、半导体、表面反应和许多其他过程中的典型过程。反应扩散系统在科学的许多分支中无处不在，几十年来一直吸引着科学家，工程师和数学家的注意力。螺旋波，空间规则，具有各种对称性（六边形，条纹等）的固定图案等迷人结构的形成。以及化学湍流使得反应扩散系统成为许多正在进行的研究的主题。虽然许多方面的异常扩散已被广泛研究，非线性动力学和图案的形成方面一直是只有非常有限的工程数量的主题。

The Principal Investigators and their colleagues develop a theory of pattern formation, nonlinear dynamics and transport in systems with anomalous diffusion, and apply it to a number of significant problems such as pattern-forming reaction-diffusion problems and drug delivery problems. Unlike regular diffusion, in which the dependence of the mean square displacement of a randomly walking particle on time is linear, in anomalous diffusion the mean square displacement behaves as a power function of time. If the exponent is less than one the diffusion process is slower than normal diffusion and is called subdiffusion, while if the exponent exceeds one it is faster than normal and is called superdiffusion. A mathematical description of anomalous diffusion involves integro-differential operators which have to be derived from appropriate continuous time random walk models, and which are difficult to study. Specifically, the investigators study (i) Pattern formation in growing domains, both for normal diffusion and anomalous diffusion, focusing on the singular perturbation case when some of the diffusion coefficients are asymptotically small; (ii) Turing pattern selection in reaction-anomalous diffusion systems, in particular, the selection of stripes or spots; (iii) Localized states in reaction-superdiffusion systems that appear due to the phenomenon of pinning of the front e.g. between the stripes and time periodic cells; and (iv) Drug delivery problems which include the development of an approximate analytic theory of subdiffusive problems with moving free boundaries; the study of models of bioerodible controlled drug delivery devices governed by subdiffusive transport of the chemicals, transdermal drug release in the presence of an electric field i.e. accompanied by iontophoresis and others.Controlled drug delivery has been attracting a great deal of attention in the medical community for years as an efficient way of providing treatment for a wide class of diseases. The common principle on which various drug delivery devices are based is mass transfer of the given drug towards particular organs, in which either the mass transfer rate, or place, or both are prescribed according to certain medical protocols. Much progress has been achieved in the design and development of various controlled drug delivery systems, and many people routinely take medicine designed for controlled release. Mathematical modeling of drug delivery systems is very important since it can provide a better understanding of and a quantitative description of the physical, chemical and biological processes governing the performance of the systems. On the basis of this description, better controlled drug delivery systems can be designed. There exists experimental evidence that drug diffusion toward the biological target is not normal but rather very slow, so-called subdiffusion, as the drug molecule has to diffuse through a very crowded environment. The investigators study drug transport governed by subdiffusion in order to obtain a better understanding of drug delivery processes. In addition, the investigators study other types of problems in which subdiffusion as well as superdiffusion (which is faster than normal diffusion) are important. This is pattern formation in reaction-diffusion systems. Subdiffusion often occurs in biogels, porous media and polymers while superdiffusion is typical of some processes in plasmas, semiconductors, surface reactions and many others. Reaction-diffusion systems are ubiquitous in many branches of science and have been attracting the attention of scientists, engineers and mathematicians for decades. The formation of such fascinating structures as spiral waves, spatially-regular, stationary patterns with various symmetries (hexagonal, stripe, etc.) as well as chemical turbulence have made reaction-diffusion systems the subject of numerous ongoing investigations. Although many aspects of anomalous diffusion have been extensively studied, nonlinear dynamic and pattern formation aspects have been the subject of only a very limited number of works.