Zymantas Darbenas

• We consider the fast reaction limit of the Keller--Rubinow model for Liesegang rings which was rigorously formulated by D. Hilhorst, R. van der Hout, M. Mimura, and I. Ohnishi in 2009; we shall refer to it as the HHMO-model in the following. Using a combination of analytical and numerical methods, we demonstrate a mechanism which suggests that the solution of this model possesses an infinite number of precipitation regions, but these regions accumulate in a finite region of space-time. This is done by introducing modifications which simplify the HHMO-model. This simplified model is shown to be equivalent to solving a fixed point functional equation of integral type. Provided that the precipitation region is followed by non-precipitation region and vice versa, we prove that all those regions accumulate at a finite point. Beyond the accumulation point, the solution can only be continued in a weak sense in which the precipitation indicator function takes fractional values and may be interpreted as a precipitation density function. We demonstrate the existence of that extended solution. Its uniqueness is shown under the assumption that the precipitation attains fractional values only. The technique involves the theory of Volterra integral equation for weakly degenerate cordial kernel functions. In separate chapter we present new results for those kernel functions. Furthermore, numerical evidence suggests that the concentration function converges, in a well-defined sense in the long-time limit, to a self-similar solution for which an explicit expression is derived. It is achieved when the precipitation functions is interchanged with self-similar scaling profile. That behaviour is proven after modifying the full HHMO-model according to numerical results, to be precise, the precipitation functions is assumed to converge strongly or weakly to the self-similar scaling profile. Moreover, we establish several uniqueness theorems for the full model.