Asymptotically Stable Stationary Solutions of the Reaction–Diffusion–Advection Equation with Discontinuous Reaction and Advection Termsстатья
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Аннотация:We study the Lyapunov asymptotic stability of the stationary solution of the spatially one-dimensional initial–boundary value problem for a nonlinear singularly perturbed differential equation of the reaction–diffusion–advection (RDA) type in the case where the advectionand reaction terms undergo a discontinuity of the first kind at some interior point of an interval.Sufficient conditions are derived for the existence of a stable stationary solution with a largegradient near the point of discontinuity. An asymptotic method of differential inequalities isused to prove the existence and stability theorems. The resulting stability conditions can beemployed to create mathematical models and develop numerical methods for solving “stiff” problems arising in various applications, for example, when simulating combustion processes and innonlinear wave theory.