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.
This one year grant was a continuation of NSF grant DMS 0707445, the purpose of which was to study the effect on pattern formation of both reaction-diffusion systems and reaction-anomalous diffusion systems, both analytically and numerically, and to compare and contrast this effect in the two systems. In total, nine papers citing the grants were published. Both reaction-diffusion and reaction-anomalous diffusion systems arise in all sorts of applications. For example, subdiffusion, which corresponds to diffusion that is slower than ordinary diffusion, often occurs, e.g., in gels, porous media and polymers, while superdiffusion, which corresponds to diffusion that is faster than ordinary diffusion is typical of some processes in plasma and turbulence, surface diffusion, diffusion in porous media as well as in geophysical, geological and other processes. Our investigations shed light on these and other applications. New type of systems behavior that involves slowly moving and stationary localized pulses was discovered. In particular, we obtained new results for the Brusselator model that has long been considered the paradigm for the study of nonlinear equations. A graduate student has been working on the project. He has already completed four published papers citing this grant and its predecessor. We have organized a research seminar on anomalous diffusion in which our graduate student is an active participant.
