Nonstationary Equations for the Reaction Layer with the Degenerate Equilibrium Points

Abstract

We consider a nonstationary process of spreading some substance in a one-dimensional spatially inhomogeneous system of cells. It is assumed that a change in the concentration of \(u_n (t)\) in a cell with the number \(n\) with time \(t\) is determined by the difference in concentration in this cell and in its two neighbors on the left and on the right, as well as the source density, which depends on \(n\) and depends on \(u_n (t)\). Such a model leads to the initial-boundary value problem for the differentialdifference equation (differentiation with respect to t variable, the difference expression with respect to \(n\)). With a sufficiently small difference in concentration in each pair of neighboring cells we can replace the difference expression by the second partial derivative with respect to the spatial coordinate, and describe the propagation by the reaction-diffusion equation. This equation belongs to the class of quasilinear parabolic equations. It is assumed that the density of the sources vanishes (with changing the sign) at three values of the concentration, two of which, lower and upper, are stable. There is also an intermediate unstable state with zero source density, in which the sign reversal also takes place. The peculiarity of our model is that we assume, that two extreme roots of the source density function are degenerate (with an integer or fractional exponent). We intend to show analytically and by the computer simulation, that this model leads to the fact, that the rate of asymptotic aspiration of concentration to equilibrium values for a moving front becomes power-law instead of exponential, which takes place for standard models. In the paper, we have constructed a formal asymptotics solution of the initialboundary value problem for the reaction-diffusion equation in a homogeneous medium with a power-law dependence of the source density on the temperature, an upper and lower solutions are constructed, a rigorous justification of the formal asymptotics is given. Precise solutions of the diffusion reaction equation are constructed for a wide class of source density functions.

Keywords

• nonlinear differential equations
• asymptotic methods
• contrast structures
• differential inequalities
• combustion