Stabilization of the front in a medium with discontinuous characteristics

N. T. Levashova ; E. A. Chunzhuk ; A. O. Orlov

Teoretičeskaâ i matematičeskaâ fizika, Tome 220 (2024) no. 1, pp. 93-112

Résumé

We study the autowave front propagation in a medium with discontinuous characteristics and the conditions for its stabilization to a stationary solution with a large gradient at the interface between media in the one-dimensional case. The asymptotic method of differential inequalities, based on constructing an asymptotic approximation of the solution, is the main method of study. We develop an algorithm for constructing such an approximation for the solution of the moving front form in a medium with discontinuous characteristics. The application of such an algorithm requires a detailed analysis of the behavior of the solution in neighborhoods of two singular points: the front localization point and the medium discontinuity point. As a result, we obtain a system of equations for the front propagation speed; this distinguishes this paper from the previously published ones. The developed algorithm can be used to describe autowave propagation in layered media. The results can also be extended to the multidimensional case.

Détail

Mots-clés : parabolic equation
Keywords: discontinuous coefficient, internal transition layer, method of differential inequalities, upper and lower solutions, asymptotically stable solution, moving front.

Bibliographie

[1] Ya. B. Zeldovich, G. I. Barenblatt, V. B. Librovich, G. M. Makhviladze, Matematicheskaya teoriya goreniya i vzryva, Nauka, M., 1980 | MR | MR

[2] R. A. Fitzhugh, “Impulses and physiological states in theoretical model of nerve membrane”, Biophys. J., 1:6 (1961), 445–466 | DOI

[3] A. E. Sidorova, N. T. Levashova, A. E. Semina, “Avtovolnovaya model morfogeneza megapolisov v predstavleniyakh neodnorodnykh aktivnykh sred”, Izv. RAN. Ser. fiz., 83:1 (2019), 106–112 | DOI

[4] N. Levashova, A. Sidorova, A. Semina, Mingkang Ni, “A spatio-temporal autowave model of shanghai territory development”, Sustainability, 11:13 (2019), 3658, 13 pp. | DOI

[5] P. C. Fife, J. B. McLeod, “The approach of solutions of nonlinear diffusion equations to travelling front solutions”, Arch. Rational Mech. Anal., 65:4 (1977), 335–361 | DOI | MR

[6] Ya. I. Kanel, “O stabilizatsii reshenii uravnenii teorii goreniya pri finitnykh nachalnykh funktsiyakh”, Matem. sb., 65:3 (1964), 398–413 | MR | Zbl

[7] A. I. Volpert, V. A. Volpert, Vl. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, 140, AMS, Providence, RI, 1994 | MR

[8] N. N. Nefedov, “Razvitie metodov asimptoticheskogo analiza perekhodnykh sloev v uravneniyakh reaktsii-diffuzii-advektsii: teoriya i primenenie”, Zh. vychisl. matem. i matem. fiz., 61:12 (2021), 2074–2094 | DOI | DOI

[9] Yu. V. Bozhevolnov, N. N. Nefedov, “Dvizhenie fronta v parabolicheskoi zadache reaktsiya-diffuziya”, Zh. vychisl. matem. i matem. fiz., 50:2 (2010), 276–285 | DOI | MR

[10] N. T. Levashova, O. A. Nikolaeva, A. D. Pashkin, “Modelirovanie raspredeleniya temperatury na granitse razdela voda-vozdukh s ispolzovaniem teorii kontrastnykh struktur”, Vestn. Mosk. un-ta. Ser. 3. Fiz. Astron., 2015, no. 5, 12–16 | DOI

[11] N. N. Nefedov, M. K. Ni, “Vnutrennie sloi v odnomernom uravnenii reaktsiya-diffuziya s razryvnym reaktivnym chlenom”, Zh. vychisl. matem. i matem. fiz., 55:12 (2015), 2042–2048 | DOI | DOI | MR

[12] N. N. Nefedov, N. T. Levashova, A. O. Orlov, “Asimptoticheskaya ustoichivost statsionarnogo resheniya s vnutrennim perekhodnym sloem zadachi reaktsiya-diffuziya s razryvnym reaktivnym slagaemym”, Vestn. Mosk. un-ta. Ser. 3. Fiz. Astron., 2018, no. 6, 3–10 | DOI

[13] N. N. Nefedov, E. I. Nikulin, A. O. Orlov, “Existence of contrast structures in a problem with discontinuous reaction and advection”, Russ. J. Math. Phys., 29:2 (2022), 214–224 | DOI | MR

[14] A. B. Vasileva, V. F. Butuzov, Asimptoticheskie metody v teorii singulyarnykh vozmuschenii, Vysshaya shkola, M., 1990 | DOI | MR

[15] V. F. Butuzov, N. N. Nefedov, K. R. Shnaider, “O formirovanii i rasprostranenii rezkikh perekhodnykh sloev v parabolicheskikh zadachakh”, Vestn. Mosk. un-ta. Ser. 3. Fiz. Astron., 2005, no. 1, 9–13 | MR

[16] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, London, 2004 | DOI | MR

[17] V. N. Pavlenko, O. V. Ulyanova, “Metod verkhnikh i nizhnikh reshenii dlya uravnenii ellipticheskogo tipa s razryvnymi nelineinostyami”, Izv. vuzov. Matem., 1998, no. 11, 69–76 | MR | Zbl

[18] V. N. Pavlenko, O. V. Ulyanova, “Metod verkhnikh i nizhnikh reshenii dlya uravnenii parabolicheskogo tipa s razryvnymi nelineinostyami”, Differents. uravneniya, 38:4 (2002), 499–504 | DOI | MR

[19] A. B. Vasileva, V. F. Butuzov, Asimptoticheskie razlozheniya reshenii singulyarno vozmuschennykh uravnenii, Nauka, M., 1973 | MR

[20] B. V. Tischenko, “Suschestvovanie, lokalnaya edinstvennost i asimptoticheskaya ustoichivost pogransloinogo resheniya kraevoi zadachi Neimana dlya sistemy dvukh nelineinykh uravnenii s raznymi stepenyami malogo parametra”, Vestn. Mosk. un-ta. Ser. 3. Fiz. Astron., 2021, no. 5, 31–37 | DOI

[21] V. F. Butuzov, N. T. Levashova, A. A. Melnikova, “Kontrastnaya struktura tipa stupenki v singulyarno vozmuschennoi sisteme uravnenii s razlichnymi stepenyami malogo parametra”, Zh. vychisl. matem. i matem. fiz., 52:11 (2012), 1983–2003 | DOI | MR

[22] K. A. Kotsyubinskii, N. T. Levashova, A. A. Melnikova, “Stabilizatsiya resheniya vida dvizhuschegosya fronta v uravnenii reaktsiya-diffuziya”, Vestn. Mosk. un-ta. Ser. 3. Fiz. Astron., 2021, no. 6, 3–11

[23] D. V. Lukyanenko, A. A. Borzunov, M. A. Shishlenin, “Solving coefficient inverse problems for nonlinear singularly perturbed equations of the reaction-diffusion-advection type with data on the position of a reaction front”, Commun. Nonlinear Sci. Numer. Simul., 99 (2021), 105824, 10 pp. | DOI | MR