Nonstationary Equations for the Reaction Layer with the Degenerate Equilibrium Points

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.

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