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Asymptotically Stable Stationary Solutions of the Reaction–Diffusion–Advection Equation with Discontinuous Reaction and Advection Terms

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Abstract

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 advection and 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 large gradient near the point of discontinuity. An asymptotic method of differential inequalities is used to prove the existence and stability theorems. The resulting stability conditions can be employed to create mathematical models and develop numerical methods for solving “stiff” problems arising in various applications, for example, when simulating combustion processes and in nonlinear wave theory.

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ACKNOWLEDGMENTS

This research was carried out at the Department of Mathematics of the Faculty of Physics of Lomonosov Moscow State University with support from Moscow Center for Fundamental and Applied Mathematics.

Funding

This work was financially supported by the Russian Science Foundation, project no. 18-11-00042.

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, 119991, Russia

    N. T. Levashova, N. N. Nefedov & O. A. Nikolaeva

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  1. N. T. Levashova
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  2. N. N. Nefedov
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  3. O. A. Nikolaeva
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Correspondence to N. T. Levashova, N. N. Nefedov or O. A. Nikolaeva.

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Translated by V. Potapchouck

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Levashova, N.T., Nefedov, N.N. & Nikolaeva, O.A. Asymptotically Stable Stationary Solutions of the Reaction–Diffusion–Advection Equation with Discontinuous Reaction and Advection Terms. Diff Equat 56, 605–620 (2020). https://doi.org/10.1134/S0012266120050067

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