Asymptotic stability of a coupled diffusion system arising from gas-liquid reactions
C. V. Pao · Rocky Mountain Journal of Mathematics · 1982
This paper is concerned with the asymptotic behavior of the time dependent solution in relation to the corresponding steady-state solution for a nonlinear coupled reaction-diffusion system arising from gas-liquid absorption.Existence and uniqueness of both time-dependent and steady-state solutions are discussed, and various boundary conditions are included in the discussion.It is shown in the case of a homogeneous system that for any non-negative initial function the time dependent solution converges exponentially to zero as t -• oo when the boundary condition is of either Dirichlet or mixed type.However, for Neumann type boundary condition, multiple constant steady-state solutions exist and the time-dependent solution may converge to any one of these steady-states.Depending on the relative magnitude between the initial functions, convergence of the time-dependent solution to one of these constant states is explicitly given.For a nonhomogeneous system with nonzero boundary or internal data the convergence of the time-dependent solutions also depends on the relative magnitude between the components of the steadystate solution.A characterization of the stability and instability of a steady-state solution is established, and in the case of stability an estimate of the stability region is given. Introduction.In the theory of a gas-liquid diffusion reaction system in a /^-dimensional medium Q the concentration of the dissolved gas u = u(t, x) and the reactant v = v(t 9 x) are governed by the coupled reaction-diffusion equations (cf.[2-4, 6, 12])where A is the Laplacian operator, D Ì9 D 2 are the diffusion coefficients, k\ 9 k 2 are the reaction rate constants and r { = -k-uv represent the rate of reactions.A more general reaction rate is given by rand is called the (m, n)th order reaction (cf.[4]).Motivated by the above