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The reaction-diffusion processes in a growing domain involves a dilution termthat modifies the properties of the homogeneous state that, in contrast to afixed domain, depends on time. We study how the dilution term changes thesteady concentrations and modifies the stability properties of theperturbations. We propose a solution for the homogeneous state thatincorporates these factors and is valid for slow variation of the size of thedomain which is based on a linear approach and has been tested againstnumerical solutions for different types of growing: exponential, linear,quadratic and oscillatory. We prove that the deviation of the steady state isproportional to the fixed point concentration, and occurs most notably forexponential growth. Systems with linear or quadratic growth tend to recover thestate that would have in absence of diffusion, whereas the oscillatoryvariation of the domain size produce temporal oscillations of theconcentration. Regarding the Turing conditions for the apparition of spatialpatterns, we study those related to stability in absence of diffusion andestablish that in a growing domain it depends upon: the change in the steadystate concentrations, the local change of volume that affects the concentrationand the stability that the reactive system would have in absence of dilution.These conditions provide richer conditions for the emergence of patterns thanthose found in a fixed domain. The formal results provided in this work areverified against numerical simulations of the homogeneous state for theBrusselator and BVAM reactions and we discuss how these variations of thehomogeneous state can give raise to crucial differences in the formation ofTuring patterns in growing domains.

Turing patterns on a two-component isotropic growing system. Part 1: Homogeneous state and stability of perturbations in absence of diffusion