Stabilization of solutions for semilinear parabolic systems as .
Systems of reaction-diffusion equations with spatially distributed hysteresis
We study systems of reaction-diffusion equations with discontinuous spatially distributed hysteresis on the right-hand side. The input of the hysteresis is given by a vector-valued function of space and time. Such systems describe hysteretic interaction of non-diffusive (bacteria, cells, etc.) and diffusive (nutrient, proteins, etc.) substances leading to formation of spatial patterns. We provide sufficient conditions under which the problem is well posed in spite of the assumed discontinuity of...
The continuum reaction-diffusion limit of a stochastic cellular growth model
A competition-diffusion system, where populations of healthy and malignant cells compete and move on a neutral matrix, is analyzed. A coupled system of degenerate nonlinear parabolic equations is derived through a scaling procedure from the microscopic, Markovian dynamics. The healthy cells move much slower than the malignant ones, such that no diffusion for their density survives in the limit. The malignant cells may locally accumulate, while for the healthy ones an exclusion rule is considered....
The dynamics of weakly interacting fronts in an adsorbate-induced phase transition model
Hildebrand et al. (1999) proposed an adsorbate-induced phase transition model. For this model, Takei et al. (2005) found several stationary and evolutionary patterns by numerical simulations. Due to bistability of the system, there appears a phase separation phenomenon and an interface separating these phases. In this paper, we introduce the equation describing the motion of two interfaces in and discuss an application. Moreover, we prove the existence of the traveling front solution which approximates...
The Feynman-Kac formula for a system of parabolic equations
The Wolff gradient bound for degenerate parabolic equations
The spatial gradient of solutions to non-homogeneous and degenerate parabolic equations of -Laplacean type can be pointwise estimated by natural Wolff potentials of the right hand side measure.
Time-dependent invariant regions for parabolic systems related to one- dimensional nonlinear elasticity
A parabolic system arisng as a viscosity regularization of the quasilinear one-dimensional telegraph equation is considered. The existence of - a priori estimates, independent of viscosity, is shown. The results are achieved by means of generalized invariant regions.
Uniform boundedness and global existence of solutions for reaction-diffusion systems with a balance law and a full matrix of diffusion coefficients.
Матрицы Грина граничных задач для параболических по И.Г. Петровскому систем общего вида
Матрицы Грина граничных задач для параболических по И.Г. Петровскому систем общего вида . II
О граничных значениях обобщенных решений систем параболических уравнений
О некоторых нелинейных системах, описывающих динамику конкурирующих биологических видов
Об единственности решения задачи Коши и асимптотической единственности для положительных решений уравнений с частными производными.
Оценки решений параболических систем в васовых классах Гёльдера и некоторые их приближения