MURPHYS-HSFS-2014 - 7th International Workshop on Multi-Rate Processes & Hysteresis, 2nd International Workshop on Hysteresis and Slow-Fast Systems, April 7-11, 2014 - Abstract

Nefedov, Nikolay

Multiple scale reaction-diffusion-advection problems with moving fronts

For some cases of initial boundary value problem for the equation beginequationlabele varepsilon^2Delta u - frac partial upartial t=f(u,nabla u, x,varepsilon),quad xin mathcal Dsubset R^N, , t>0, endequation which plays important role in many applications and is called reaction-diffusion-advection equation we state the conditions which imply the existence of solutions with moving internal layers - fronts. Particularly we consider the following cases of equation (refe): 1. Reaction-diffusion equation beginequation* varepsilon^2 fracpartial^2 upartial x^2 - frac partial upartial t=f(u, x,t,varepsilon),quad xin (0,1), t>0. endequation* 2. Reaction-advection-diffusion equation beginequation* varepsilon fracpartial^2 upartial x^2 - A(u,x)frac partial upartial x - frac partial upartial t=f(u, x,t,varepsilon),quad xin (0,1), t>0. endequation* For the initial boundary value problems for these equation we prove the existence of fronts and give its asymptotic approximation. We also give an extension of our results for the reaction-diffusion system beginalign* beginsplit &varepsilon^q fracpartial^2 upartial x^2 - frac partial vpartial t=f(u,v, x,varepsilon),
&varepsilon^p fracpartial^2 upartial x^2 -fracpartial upartial t = g(u,v,x,varepsilon),quad xin (0,1), t>0. endsplit endalign* Our investigations are based on asymptotic method of differential inequalities and use so-called positivity property of the operators producing formal asymptotics. The results of the work is a further development of the results of papers [1] -[5]. This work is supported by RFBR, pr. N 13-01-00200. beginitemize item[[1]] emphVasilieva A.B., Butuzov V.F., Nefedov N.N., “Contrast structures in singularly perturbed problems” (in russian), Fundamentalnaja i prikladnala matematika, 4, No. 3, 799-851 (1998). item[[2]] emph Nefedov N.N., The Method of Differential Inequalities for Some Classes of Nonlinear Singularly Perturbed Problems with Internal Layers, it Differ. Uravn., 1995. vol. 31. no. 7. pp. 1142-1149. item[[3]] emphA.B. Vasileva, V.F. Butuzov, N.N. Nefedov.Singularly Perturbed problems with Boundary and Internal Layers. Proceedings of the Steklov Institute of Mathmatics, 2010, Vol.268, pp.258-273. item[[4]] emphN. N. Nefedov, A. G. Nikitin, M. A. Petrova, and L. Recke, Moving fronts in integro-parabolic reaction-advection-diffusion equations, Diff. Equations bf 47, 1318--1332 (2011). item[[5]] emph N. N. Nefedov, L. Recke, and K. R. Schneider, Existence and asymptotic stability of periodic solutions with an interior layer of reaction-advection-diffusion equations, J. Math. Anal. Appl. bf 405, 90--103 (2013). enditemize