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Abstract
This paper is devoted to the investigation of ill-posed boundary-value problem for system of parabolic type equations with changing time direction with two degenerate lines. The problem under consideration is ill-posed in the sense of J. Hadamard, namely, there is no continuous dependence of the solution on the initial data. Such equations have many different applications, for example, describe the processes of heat propagation in inhomogeneous media, the interaction of filtration flows, mass transfer near the surface of an aircraft, and the description of complex viscous fluid flows. As possible applications should also indicate the task of calculating heat exchangers, in which the counter flow principle is used. Theorems on the uniqueness and conditional stability of a solution on a set of well-posedness are proved. We construct a sequence of approximate (regularized) solutions that are stable on the set of well-posedness
Highlights
Introduction and preliminariesWe consider the problem of finding a solution (u(x, y, t), ν(x, y, t)) of the system of equations ( ∂ (∂t ∂ − sign(x) ∂x2) ∂ − sign(y) ∂y2 ) ν(x, y, t) = 0, ∂∂t − sign(x) ∂x2 − sign(y) ∂y2 u(x, y, t) = ν(x, y, t) (1)in the region Ω = {(x, y, t)| (−1 < x < 1) × (−1 < y < 1) × (0 < t < T < ∞), x= 0, y= 0} that satisfy the following conditions: initial conditions∏ u(x, y, 0) = f (x, y), ν(x, y, 0) = φ(x, y), (x, y) ∈ = { −1 x 1, −1 y 1}, (2)K.S
From further research it became clear that parabolic equations with a changing time direction can be considered as a special case of equations of a mixed type
The conditional correctness of problem (1)–(4) is investigate, namely, theorems on uniqueness and conditional stability were proved, and approximate solutions that are stable on the set of well-posedness are constructed
Summary
We consider the problem of finding a solution (u(x, y, t), ν(x, y, t)) of the system of equations. Hadamard) boundary-value problem for a system of parabolic type equations with a changing time direction. For parabolic type equations with changing time directions problems were first considered by M. From further research it became clear that parabolic equations with a changing time direction can be considered as a special case of equations of a mixed type. A great contribution to the study of boundary value problems for equations of mixed type was made by A. Ill-posed problems for such equations was considered in the works of H. In the previous works were examined correct and incorrect problems for the classic and mixed-type equations with one degenerate lines. The conditional correctness of problem (1)–(4) is investigate, namely, theorems on uniqueness and conditional stability were proved, and approximate solutions that are stable on the set of well-posedness are constructed
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