On the number of stationary patterns in reaction-diffusion systems. (English) Zbl 1363.35179

Brandts, J. (ed.) et al., Proceedings of the international conference ‘Applications of mathematics’, Prague, Czech Republic, November 18–21, 2015. In honor of the birthday anniversaries of Ivo Babuška (90), Milan Práger (85), and Emil Vitásek (85). Prague: Czech Academy of Sciences, Institute of Mathematics (ISBN 978-80-85823-65-3). 206-216 (2015).
The authors study a particular system of two nonlinear reaction-diffusion partial differential equations exhibiting diffusion driven instability which often leads to stationary solutions known as Turing patterns. Such systems can be used to model certain phenomena in biology, chemistry and other fields. A particular system studied in the paper leads to solutions having spatial pattern with certain number of “spots” in the computational domain. A typical property of studied systems is non-uniqueness of the solution. Interestingly, for different initial conditions, one obtain solutions that are clearly different from mathematical point of view, but “qualitatively” similar, e.g., have similar (but not always the same) number of spots with similar sizes and in similar distances. As there is no available theory, the authors approach the problem numerically, solve it for many different initial conditions and classify obtained results. Solutions that differ only by a symmetry or shift are put into one group. An interesting result of this paper is that the authors obtained surprisingly many distinct solution classes. Even though the results (as the authors admit) cannot be considered absolutely rigorous, since it is not clear that all studied solutions really converged to a steady-state, the conclusions offer interesting insight into a class of systems with very complex behavior.
For the entire collection see [Zbl 1329.00187].
Reviewer: Pavel Kůs (Praha)


35K57 Reaction-diffusion equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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