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

首席研究人员和他们的同事发展了具有反常扩散的系统中的模式形成、非线性动力学和输运理论，并将其应用于一些重要问题，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与时间的关系是线性的，而在反常扩散中，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，称为次扩散；如果指数超过1，则比正常快，称为超扩散。反常扩散的数学描述涉及积分-微分算子，它必须从适当的连续时间随机游走模型导出，并且很难研究。具体地说，研究人员研究了(I)增长区域中的图案形成，包括正常扩散和反常扩散，重点关注当一些扩散系数渐近小时的奇异扰动情况；(Ii)反应反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)反应超扩散系统中由于前沿钉扎现象而出现的局域态，例如条纹和时间周期细胞之间；以及(Iv)药物输送问题，其中包括发展具有移动自由边界的次扩散问题的近似解析理论；生物可侵蚀药物控释装置模型的研究主要受化学物质的亚扩散传输、电场作用下的经皮药物释放、离子导入等控制。多年来，药物控制给药作为一种治疗多种疾病的有效方法引起了医学界的极大关注。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转移，其中质量转移速率或位置或两者都是根据特定的医学方案规定的。在设计和开发各种受控给药系统方面取得了很大进展，许多人经常服用为控制释放而设计的药物。药物输送系统的数学模型非常重要，因为它可以更好地理解和定量描述控制系统性能的物理、化学和生物过程。在此描述的基础上，可以设计更好的受控药物输送系统。有实验证据表明，药物向生物靶点的扩散不是正常的，而是非常缓慢的，即所谓的亚扩散，因为药物分子必须在非常拥挤的环境中扩散。为了更好地了解药物的传递过程，研究人员研究了亚扩散控制的药物转运。此外，研究人员还研究了其他类型的问题，在这些问题中，次扩散和超扩散(比正常扩散更快)是重要的。这是反应扩散系统中的图案形成。二次扩散通常发生在生物凝胶、多孔介质和聚合物中，而超扩散是等离子体、半导体、表面反应和许多其他过程中的典型过程。反应扩散系统在许多科学分支中无处不在，几十年来一直吸引着科学家、工程师和数学家的注意。螺旋波、空间规则的、具有各种对称性(六边形、条纹等)的静止图案等迷人结构的形成以及化学湍流，使反应扩散系统成为许多正在进行的研究的主题。虽然反常扩散的许多方面已经得到了广泛的研究，但非线性动力学和模式形成方面的研究成果非常有限。

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.