Well-posedness of parabolic equations containing hysteresis with diffusive thresholds
- 49 Downloads
We study complex systems arising, in particular, in population dynamics, developmental biology, and bacterial metabolic processes, in which each individual element obeys a relatively simple hysteresis law (a non-ideal relay). Assuming that hysteresis thresholds fluctuate, we consider the arising reaction-diffusion system. In this case, the spatial variable corresponds to the hysteresis threshold. We describe the collective behavior of such a system in terms of the Preisach operator with time-dependent measure which is a part of the solution for the whole system. We prove the well-posedness of the system and discuss the long-term behavior of solutions.
KeywordsLactose Parabolic Equation STEKLOV Institute Neumann Boundary Condition Switching Threshold
Unable to display preview. Download preview PDF.
- 6.M. Cohn and K. Horbita, “Inhibition by glucose of the induced synthesis of the β-galactoside-enzyme system of Escherichia coli. Analysis of maintenance,” J. Bacteriol. 78, 601–612 (1959).Google Scholar
- 7.M. Cohn and K. Horbita, “Analysis of the differentiation and of the heterogeneity within a population of Eschericia coli undergoing induced β-galactosidase synthesis,” J. Bacteriol. 78, 613–623 (1959).Google Scholar
- 8.M. Delbrück, “Discussion,” in Unités biologiques douées de continuité génétique (CNRS, Paris, 1949), pp. 33–35.Google Scholar
- 10.G. Friedman, P. Gurevich, S. McCarthy, and D. Rachinskii, “Switching behaviour of two-phenotype bacteria in varying environment,” J. Phys.: Conf. Ser. (in press).Google Scholar
- 15.J. Kopfová, “Hysteresis in biological models,” J. Phys.: Conf. Ser. 55, 130–134 (2006).Google Scholar
- 18.O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasi-linear Equations of Parabolic Type (Nauka, Moscow, 1967; Am. Math. Soc., Providence, RI, 1968).Google Scholar
- 19.V. P. Mikhailov, Partial Differential Equations (Nauka, Moscow, 1983) [in Russian].Google Scholar
- 29.Ö. Winge and C. Roberts, “Inheritance of enzymatic characters in yeasts, and the phenomenon of long-term adaptation,” C. R. Trav. Lab. Carlsberg, Sér. Physiol. 24, 263–315 (1948).Google Scholar