Abstract
The work is aimed to study front solutions of a nonlinear system of parabolic equations in a two-dimensional region. The system can be considered as a mathematical model describing an abrupt change in physical characteristics of spatially heterogeneous media. We consider a system with small parameters raised to the different powers at a differential operator, that represents the difference of typical processes speeds for the system components. The study of the system is conducted by using the contrast structures theory methods, which allowed us to obtain conditions for the existence of front solutions contained in the neighborhood of a closed curve, to determine the front velocity depending on time and coordinate along the front curve, and to obtain the zero-order and the first-order terms of the asymptotic approximation to the solution. The scope of the system includes the description of autowave solutions in the field of ecology, biophysics, combustion physics and chemical kinetics. The approximate solution allows us to choose the model parameters so that the result corresponds to the processes observed, to explain and describe the characteristics of the solutions with sharp gradients, to create models with stable solutions and thereby to simplify the numerical analysis. Note that the numerical experiment for the two-dimensional spatial models requires a considerable amount of processing power and the use of parallel computing techniques and does not allow to effectively analyze and modify the model. In this paper, we obtain the asymptotic approximation that is to be justified, which can be done by the method of differential inequalities.
Highlights
The system can be considered as a mathematical model describing an abrupt change in physical characteristics
We consider a system with small parameters raised to the different powers at a differential operator
that represents the difference of typical processes speeds for the system components
Summary
С краевыми условиями Неймана на границе области и с периодическими условиями по переменной t:. Уравнение f (u, v, x, t, 0) = 0 при (v, x, t) ∈ Iv × D × (0; +∞) имеет относительно u ровно три корня u = φ(−)(v, x, t), u = φ(+)(v, x, t), u = φ0(v, x, t), такие, что φ(−)(v, x, t) < φ0(v, x, t) < φ(+)(v, x, t) и fu(φ(±)(v, x, t), v, x, t, 0) > 0, fu(φ0(v, x), v, x, t, 0) < 0. Каждое из уравнений h(±)(v, x, t) := g(φ(±)(v, x, t), v, x, t, 0) = 0, при x ∈ Dимеет относительно v единственное решение v = v(±)(x, t) ∈ Iv, причем во всей области (x, t) ∈ D × (0; +∞) выполнены неравенства v(−)(x, t) < v(+)(x, t) и hv(±)(v(±)(x, t), x, t) > 0. Алгоритм получения асимптотики следующих порядков аналогичен алгоритму для первого порядка
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