Existence of a~solution in the form of a~moving front of a~reaction-diffusion-advection problem

N. T. Levashova ; N. N. Nefedov ; A. V. Yagremtsev

Izvestiya. Mathematics , Tome 82 (2018) no. 5, pp. 984-1005.

Résumé

We consider the initial-boundary value problem for an equation of reaction-diffusion-advection type in the case when the condition of balanced advection is satisfied. We give an algorithm for constructing an asymptotic representation of a solution which has the form of a moving front, obtain the equation of motion for the point of localization of the front, and prove the existence of that solution. The proof uses the asymptotic method of differential inequalities.

Détail

Keywords: small parameter, asymptotic methods, internal transition layer, differential inequalities.
Mots-clés : equation of reaction-diffusion-advection type, motion of a front

Bibliographie

[1] N. T. Levashova, J. V. Muhartova, M. A. Davydova, N. E. Shapkina, A. V. Oltchev, “The application of the theory of contrast structures for describing wind field in spatially heterogeneous vegetation cover”, Moscow Univ. Phys. Bull., 70:3 (2015), 167–174 | MR

[2] N. T. Levashova, Yu. V. Mukhartova, A. V. Olchev, “Trekhmernoe modelirovanie turbulentnogo perenosa v prizemnom sloe atmosfery s primeneniem teorii kontrastnykh struktur”, Kompyuternye issledovaniya i modelirovanie, 8:2 (2016), 355–367

[3] N. E. Grachev, A. V. Dmitriev, D. S. Senin, V. T. Volkov, N. N. Nefedov, “Modelirovanie dinamiki fronta vnutriplastovogo goreniya”, Vych. met. programmirovanie, 11:4 (2010), 306–312

[4] V. T. Volkov, N. N. Nefedov, N. E. Grachev, D. S. Senin, “Otsenka parametrov fronta vnutriplastovogo goreniya pri zakachke vozdukha v neftyanoi plast”, Neftyanoe khozyaistvo, 2010, no. 4, 93–96

[5] A. B. Vasil'eva, V. F. Butuzov, N. N. Nefedov, “Singularly perturbed problems with boundary and internal layers”, Proc. Steklov Inst. Math., 268 (2010), 258–273 | DOI | MR | Zbl

[6] N. T. Levashova, N. N. Nefedov, A. V. Yagremtsev, “Contrast structures in the reaction-diffusion-advection equations in the case of balanced advection”, Comput. Math. Math. Phys., 53:3 (2013), 273–283 | DOI | DOI | MR | Zbl

[7] N. N. Nefedov, “An asymptotic method of differential inequalities for the investigation of periodic contrast structures: existence, asymptotics, and stability”, Differ. Equ., 36:2 (2000), 298–305 | DOI | MR | Zbl

[8] V. T. Volkov, N. N. Nefedov, “Development of the asymptotic method of differential inequalities for investigation of periodic contrast structures in reaction-diffusion equations”, Comput. Math. Math. Phys., 46:4 (2006), 585–593 | DOI | MR | Zbl

[9] Yu. V. Bozhevol'nov, N. N. Nefëdov, “Front motion in the parabolic reaction-diffusion problem”, Comput. Math. Math. Phys., 50:2 (2010), 264–273 | DOI | MR | Zbl

[10] V. Volkov, N. Nefedov, “Asymptotic-numerical investigation of generation and motion of fronts in phase transition models”, Numerical analysis and its applications, Lecture Notes in Comput. Sci., 8236, Springer, Heidelberg, 2013, 524–531 | DOI | MR | Zbl

[11] E. A. Antipov, N. T. Levashova, N. N. Nefedov, “Asymptotics of the front motion in the reaction-diffusion-advection problem”, Comput. Math. Math. Phys., 54:10 (2014), 1536–1549 | DOI | DOI | MR | Zbl

[12] P. Hess, Periodic-parabolic boundary value problems and positivity, Pitman Res. Notes Math. Ser., 247, Longman Scientific Technical, Harlow; John Wiley Sons, Inc., New York, 1991, viii+139 pp. | MR | Zbl

[13] D. H. Sattinger, “Monotone methods in nonlinear elliptic and parabolic boundary value problems”, Indiana Univ. Math. J., 21:11 (1972), 979–1000 | DOI | MR | Zbl

[14] N. N. Nefedov, “The method of differential inequalities for some classes of nonlinear singularly perturbed problems with internal layers”, Differ. Equ., 31:7 (1995), 1077–1085 | MR | Zbl

[15] N. Nefedov, “Comparison principle for reaction-diffusion-advection problems with boundary and internal layers”, Numerical analysis and its applications, Lecture Notes in Comput. Sci., 8236, Springer, Heidelberg, 2013, 62–72 | DOI | MR | Zbl

[16] N. N. Nefedov, L. Recke, K. R. Schneider, “Existence and asymptotic stability of periodic solutions with an interior layer of reaction-advection-diffusion equations”, J. Math. Anal. Appl., 405:1 (2013), 90–103 | DOI | MR | Zbl

[17] N. N. Nefedov, K. Sakamoto, “Multi-dimensional stationary internal layers for spatially inhomogeneous reaction-diffusion equations with balanced nonlinearity”, Hiroshima Math. J., 33:3 (2003), 391–432 | MR | Zbl

[18] Xinfu Chen, “Generation and propagation of interfaces for reaction–diffusion equations”, J. Differential Equations, 96:1 (1992), 116–141 | DOI | MR | Zbl

[19] C. Mantegazza, Lecture notes on mean curvature flow, Progr. Math., 290, Birkhäuser/Springer Basel AG, Basel, 2011, xii+166 pp. | DOI | MR | Zbl

[20] N. N. Nefedov, E. I. Nikulin, “Existence and stability of periodic contrast structures in the reaction-advection-diffusion problem in the case of a balanced nonlinearity”, Differ. Equ., 53:4 (2017), 516–529 | DOI | DOI | MR | Zbl

[21] A. I. Volpert, Vit. A. Volpert, Vl. A. Volpert, Traveling wave solutions of parabolic systems, Transl. from the Russian manuscript, Transl. Math. Monogr., 140, 2nd ed., Amer. Math. Soc., Providence, RI, 2000, 448 pp. | MR | Zbl

[22] M. A. Davydova, S. A. Zakharova, N. T. Levashova, “On one model problem for the reaction-diffusion-advection equation”, Comput. Math. Math. Phys., 57:9 (2017), 1528–1539 | DOI | DOI | MR | Zbl

[23] D. V. Lukyanenko, V. T. Volkov, N. N. Nefedov, L. Recke, K. Schneider, “Analytic-numerical approach to solving singularly perturbed parabolic equations with the use of dynamic adapted meshes”, Model. i analiz inform. sistem, 23:3 (2016), 334–341 | DOI | MR