Abstract
Investigations of initial boundary value problems for parabolic equations solutions are an important component of mathematical modeling. In this regard of special interest for mathematical modeling are the boundary value problem solutions that undergo sharp changes in any area of space. Such areas are called internal transitional layers. In case when the position of a transitional layer changes over time, the solution of a parabolic equation behaves as a moving front. For the purpose of proving the existence of such initial boundary value problem solutions, the method of differential inequalities is very effective. According to this method the so-called upper and lower solutions are to be constructed for the initial boundary value problem. The essence of an asymptotic method of differential inequalities is in receiving the upper and lower solutions as modifications of asymptotic submissions of the solutions of boundary value problems. The existence of the upper and lower solutions is a sufficient condition of existence of a solution of a boundary value problem. While proving the differential inequalities the so-called ”quasimonotony” condition is essential. In the present work it is considered how to construct the upper and lower solutions for the system of the parabolic equations under various conditions of quasimonotony.
Highlights
Investigations of initial boundary value problems for parabolic equations solutions are an important component of mathematical modeling
lower solutions are to be constructed for the initial boundary value problem
lower solutions is a sufficient condition of existence of a solution
Summary
Асимптотическое представление решения задачи (1) будем строить, считая, что выполняются следующие условия: Условие А1. Уравнение f (u, v, x, 0) = 0 имеет относительно u единственное решение u = φ(v, x) ∈ Iu, причем fu(φ(v, x), v, x, 0) > 0. Асимптотическое представление решения задачи (1) в каждый момент времени строится отдельно в каждой из этих подобластей: u(−), (x, t) ∈ D −, u = u(+), (x, t) ∈ D +; v(−), (x, t) ∈ D −, v = v(+), (x, t) ∈ D +. Для каждого значения x∗(t) при фиксированном значении t ∈ (0; T ] существует единственная величина W такая, что задача (4) имеет единственное решение v(ξ, x∗). При выполнении условий А.1 – А.4, используя алгоритм Васильевой, можно получить функции Un, Vn асимптотическое представление произвольного порядка n решения задачи (1), а также разложение (3) порядка n
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