TY - JOUR

T1 - The propagation phenomenon of solutions of a parabolic problem on the sphere

AU - Kazmierczak, Bogdan

AU - Tsai, Je Chiang

AU - Bialecki, Slawomir

N1 - Publisher Copyright:
© 2018 World Scientific Publishing Company.

PY - 2018/9/1

Y1 - 2018/9/1

N2 - In this paper, we study propagation phenomena on the sphere using the bistable reaction-diffusion formulation. This study is motivated by the propagation of waves of calcium concentrations observed on the surface of oocytes, and the propagation of waves of kinase concentrations on the B-cell membrane in the immune system. To this end, we first study the existence and uniqueness of mild solutions for a parabolic initial-boundary value problem on the sphere with discontinuous bistable nonlinearities. Due to the discontinuous nature of reaction kinetics, the standard theories cannot be applied to the underlying equation to obtain the existence of solutions. To overcome this difficulty, we give uniform estimates on the Legendre coefficients of the composition function of the reaction kinetics function and the solution, and a priori estimates on the solution, and then, through the iteration scheme, we can deduce the existence and related properties of solutions. In particular, we prove that the constructed solutions are of C2,1 class everywhere away from the discontinuity point of the reaction term. Next, we apply this existence result to study the propagation phenomenon on the sphere. Specifically, we use stationary solutions and their variants to construct a pair of time-dependent super/sub-solutions with different moving speeds. When applied to the case of sufficiently small diffusivity, this allows us to infer that if the initial concentration of the species is above the inhomogeneous steady state, then the species will exhibit the propagating behavior.

AB - In this paper, we study propagation phenomena on the sphere using the bistable reaction-diffusion formulation. This study is motivated by the propagation of waves of calcium concentrations observed on the surface of oocytes, and the propagation of waves of kinase concentrations on the B-cell membrane in the immune system. To this end, we first study the existence and uniqueness of mild solutions for a parabolic initial-boundary value problem on the sphere with discontinuous bistable nonlinearities. Due to the discontinuous nature of reaction kinetics, the standard theories cannot be applied to the underlying equation to obtain the existence of solutions. To overcome this difficulty, we give uniform estimates on the Legendre coefficients of the composition function of the reaction kinetics function and the solution, and a priori estimates on the solution, and then, through the iteration scheme, we can deduce the existence and related properties of solutions. In particular, we prove that the constructed solutions are of C2,1 class everywhere away from the discontinuity point of the reaction term. Next, we apply this existence result to study the propagation phenomenon on the sphere. Specifically, we use stationary solutions and their variants to construct a pair of time-dependent super/sub-solutions with different moving speeds. When applied to the case of sufficiently small diffusivity, this allows us to infer that if the initial concentration of the species is above the inhomogeneous steady state, then the species will exhibit the propagating behavior.

KW - Propagation

KW - bistable kinetics

KW - sphere

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U2 - 10.1142/S0218202518500483

DO - 10.1142/S0218202518500483

M3 - Article

AN - SCOPUS:85049593528

SN - 0218-2025

VL - 28

SP - 2001

EP - 2067

JO - Mathematical Models and Methods in Applied Sciences

JF - Mathematical Models and Methods in Applied Sciences

IS - 10

ER -