Regularity of solutions to a reaction–diffusion equation on the sphere

the Legendre series approach

Slawomir Bialecki, Bogdan Kaźmierczak, Dominika Nowicka, Je-Chiang Tsai

Research output: Contribution to journalArticle

1 Citation (Scopus)

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.

Original languageEnglish
Pages (from-to)5349-5369
Number of pages21
JournalMathematical Methods in the Applied Sciences
Volume40
Issue number14
DOIs
Publication statusPublished - 2017 Sep 30

Fingerprint

Regularity of Solutions
Mild Solution
Legendre
Reaction-diffusion Equations
Pole
Poles
Euclidean plane
Legendre polynomial
Series
Polynomials
Solution Existence
Reaction Kinetics
Oocyte
Fertilization
Cell
Infinite series
Symmetric Functions
Operator
Iterative methods
A Priori Estimates

Keywords

  • discontinuous reaction term
  • sphere
  • stationary fronts

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)

Cite this

Regularity of solutions to a reaction–diffusion equation on the sphere : the Legendre series approach. / Bialecki, Slawomir; Kaźmierczak, Bogdan; Nowicka, Dominika; Tsai, Je-Chiang.

In: Mathematical Methods in the Applied Sciences, Vol. 40, No. 14, 30.09.2017, p. 5349-5369.

Research output: Contribution to journalArticle

Bialecki, Slawomir ; Kaźmierczak, Bogdan ; Nowicka, Dominika ; Tsai, Je-Chiang. / Regularity of solutions to a reaction–diffusion equation on the sphere : the Legendre series approach. In: Mathematical Methods in the Applied Sciences. 2017 ; Vol. 40, No. 14. pp. 5349-5369.
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