Sławomir Białecki, M.Sc., Eng.

Department of Biosystems and Soft Matter (ZBiMM)
Division of Modelling in Biology and Medicine (PMBM)
position: specialist
telephone: (+48) 22 826 12 81 ext.: 448, 450
room: 321
e-mail: sbialeck

Recent publications
1.Białecki S., Kaźmierczak B., Nowicka D., Tsai J.-C., Regularity of solutions to a reaction–diffusion equation on the sphere: the Legendre series approach, MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 0170-4214, DOI: 10.1002/mma.4390, pp.1-21, 2017
Abstract:

In the paper, we study some ‘a priori’ properties of mild solutions to a single reaction–diffusion equation with discontinuous nonlinear reaction term on the two-dimensional sphere close to its poles. This equation is the counterpart of the well-studied bistable reaction–diffusion equation on the Euclidean plane. The investigation of this equation on the sphere is mainly motivated by the phenomenon of the fertilization of oocytes or recent studies of wave propagation in a model of immune cells activation, in which the cell is modeled by a ball. Because of the discontinuous nature of reaction kinetics, the standard theory cannot guarantee the solution existence and its smoothness properties. Moreover, the singular nature of the diffusion operator near the north/south poles makes the analysis more involved. Unlike the case in the Euclidean plane, the (axially symmetric) Green's function for the heat operator on the sphere can only be represented by an infinite series of the Legendre polynomials. Our approach is to consider a formal series in Legendre polynomials obtained by assuming that the mild solution exists. We show that the solution to the equation subject to the Neumann boundary condition is C1 smooth in the spatial variable up to the north/south poles and Hölder continuous with respect to the time variable. Our results provide also a sort of ‘a priori’ estimates, which can be used in the existence proofs of mild solutions, for example, by means of the iterative methods.

Keywords:

discontinuous reaction term, stationary fronts, sphere

Affiliations:
Białecki S.-IPPT PAN
Kaźmierczak B.-IPPT PAN
Nowicka D.-IPPT PAN
Tsai J.-C.-National Chung Cheng University (TW)
2.Białecki S., Kaźmierczak B., Tsai J-C., Stationary waves on the sphere, SIAM JOURNAL ON APPLIED MATHEMATICS, ISSN: 0036-1399, DOI: 10.1137/140999384, Vol.75, No.4, pp.1761-1788, 2015
Abstract:

In this paper, we investigate stationary waves on the sphere using the bistable reaction-diffusion system. The motivation of this study arises from the study of activation waves of B cells in immune systems. We analytically establish (i) the existence and uniqueness of stationary waves; (ii) the limiting wave profile for small diffusivity of diffusing species; and (iii) the stability of the constructed stationary waves. The stability result may suggest the critical role of stationary waves in the determination of initial data for initiating propagating waves on the sphere, which is consistent with the numerical results for the B-cell activation model.

Keywords:

stationary wave, sphere, bistable kinetics

Affiliations:
Białecki S.-IPPT PAN
Kaźmierczak B.-IPPT PAN
Tsai J-C.-National Chung Cheng University (TW)