External boundary effects on simultaneous diffusion and reaction processes

Abstract

External boundaries influence the spatial and temporal structure of evolution of dynamic systems governed by reaction-diffusion equations. Critical limits, i.e. thresholds for explosive growth or onset of diffusion dominated decay, are found to be caused by the presence of the boundary and to depend on: (i) the position of the boundary, where the density is assumed to be zero at any instant of time; (ii) the mutual weights (coefficients) and powers of the nonlinear reaction and diffusion processes; (iii) the initial spatial distributionHowever, for particular relations between the nonlinear powers of the reaction and diffusion terms (p = δ + 1) the critical limits do not depend on the initial conditions. In those cases (p = δ + 1) the spatial distributions tend, asymptotically in time, towards solutions for which the quantity np, where n denotes the density, satisfies a linear differential equation.The results are obtained by simulation experiments for one, two and three dimensions.Trends in the dynamic evolution of the system with an external boundary imposed are compared with the corresponding analytic results obtained for a free boundary.Interesting applications are found in various areas, e.g. in the field of high temperature fusion plasmas where the evolution of the temperature profile for the so-called H-mode (constant plasma density) is described.