Geodesic
Solution with an inner transition layer of a~two-dimensional boundary value reaction--diffusion--advection problem with discontinuous reaction and advection terms
Teoretičeskaâ i matematičeskaâ fizika, Tome 207 (2021) no. 2, pp. 293-309

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We study the problem of the existence and asymptotic stability of a stationary solution of an initial boundary value problem for the reaction–diffusion–advection equation assuming that the reaction and advection terms are comparable in size and have a jump along a smooth curve located inside the studied domain. The problem solution has a large gradient in a neighborhood of this curve. We prove theorems on the existence, asymptotic uniqueness, and Lyapunov asymptotic stability for such solutions using the method of upper and lower solutions. To obtain the upper and lower solutions, we use the asymptotic method of differential inequalities that consists in constructing them as modified asymptotic approximations in a small parameter of solutions of these problems. We construct the asymptotic approximation of a solution using a modified Vasil'eva method.
Mots-clés : reaction–diffusion–advection equation
Keywords: discontinuous term, method of differential inequalities, upper solution, lower solution, inner transition layer, small parameter. 
@article{TMF_2021_207_2_a8,
     author = {N. T. Levashova and N. N. Nefedov and O. A. Nikolaeva},
     title = {Solution with an inner transition layer of a~two-dimensional boundary value reaction--diffusion--advection problem with discontinuous reaction and advection terms},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {293--309},
     publisher = {mathdoc},
     volume = {207},
     number = {2},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2021_207_2_a8/}
}
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N. T. Levashova; N. N. Nefedov; O. A. Nikolaeva. Solution with an inner transition layer of a~two-dimensional boundary value reaction--diffusion--advection problem with discontinuous reaction and advection terms. Teoretičeskaâ i matematičeskaâ fizika, Tome 207 (2021) no. 2, pp. 293-309. http://geodesic.mathdoc.fr/item/TMF_2021_207_2_a8/