A System of Singularly Perturbed Parabolic Equations with a Power Boundary Layer

The work is devoted to the construction of the asymptotics of the solution of a singularly perturbed system of equations of parabolic type, in the case when the limit equation has a regular singularity as the small parameter tends to zero. The asymptotics of the solution of such problems contains, along with parabolic boundary layer functions, and power boundary layer functions.

The papers [16,23] are devoted to systems of singularly perturbed parabolic equations. In [16], the method of boundary functions, and in [23], the method of regularization. For singularly perturbed problems, equations are studied when the spatial derivative is preceded by a scalar function, and [24] is a matrix. In the latter case, the structure of the solution asymptotic is greatly multiplied. The order of the equation does not go down.
Problems with power boundary layers i.e. problems where, when a small parameter tends to zero, it acquires a regular feature was studied in [9,10,26]. Ordinary differential equations with a power boundary layer are studied in [8][9][10]26]. For equations of parabolic type, when a small parameter does not enter the factors of the spatial derivative, the asymptotic of the power boundary layer is constructed. In contrast to [9], in our equation there is a small parameter in front of the spatial derivative and we will improve the algorithm for constructing the asymptotic.

REGULARIZATION OF PROBLEMS
We introduce regularizing variables Hence, on the basis of (2), we find then, according to (1), we set the extended problem The notation is entered here Note that the identity holds solutions of problem (3) will be defined as then for the coefficients of this series we obtain the following iterative problems:

SOLVABILITY OF ITERATIVE PROBLEMS
We introduce the space of functions in which the iterative problems will be solved: Here We calculate the action of the operators included in the extended equation on the function u k (M ) ∈ M , for which we first decompose ψ i (t) = n j=1 α ij (t)ψ j (t), or we write by entering the notation Iterative equations write in the form Theorem 1. Let conditions 1), 2) and h k (M ) ∈ U 2 hold. Then equation (9) is solvable in U .
Function (7) will be the solution of equation (9) in U . If the functions Y k (N l ), z k (N l ) are the solution of the equations These equations are obtained by substituting (7) into equation (9), taking into account calculations (8) and representation (10). Equations (11), with appropriate boundary conditions, have solutions that satisfy the estimates [25]: . P Theorem 2. Let conditions 1), 2) and h k,3 (x, 0) = 0 hold. Then problem has a unique solution. Hereinafter, c denotes a matrix with nonzero diagonal, and c with zero diagonal elements, i.e. c = c + c. Proof. We write the relation (12) in the coordinate form Removing the degeneracy of this equation, assuming that Equating the coefficients at ψ, we get Equation (15), by virtue of condition h k ii (x, 0) = 0, under the initial condition (13), and equation (16) with condition (14) have one-to-one solutions. P

Remarks.
In iterative problems, the condition h k ii (x, 0) = 0 will be provided by the choice of the function P k i (x). Proof. Let satisfy the function (7) to the boundary conditions a): Ensuring the solvability of equation (9) assuming that Equations (18) under the initial condition (17), on the basis of Theorem 2, are uniquely solvable. Equations (19) are solved without an initial condition and have a bounded solution [27]. Equations (11) with boundary conditions (17) have solutions that can be represented as where erf c(x) = 2π −1/2 ∞ x e −t 2 dt is integral of the additional function and describes a parabolic boundary layer, d k,l (x, t), W k,l (x, t) are arbitrary functions that are chosen, like solving the equations (20) with boundary conditions from (17). Equations (20) are obtained by satisfying condition c) and taking into account that the functions erf c[ξ l /(2τ 1/2 )] and I l (ξ l , τ) have single estimates.
Thus, a unique solution to equation (9) is obtained that satisfies conditions a) − c). P

CONSTRUCTION OF SOLUTIONS OF ITERATIVE EQUATIONS
For ν = 0, 1 the equations for u ν (M ) are homogeneous; therefore, the condition of Theorem 1 holds; therefore, the solution of these equations exists and can be represented in the form (7).
The following iterative equation, on the grounds (8), has a free term We decompose f (x, t) by the system ψ i (t) and substitute it into the previous relation. Further, providing the conditions of Theorem 1, we set The first system has a smooth solution, the second system by Theorem 2 is solvable if Problem (21) is uniquely solvable, and problem (22) has a trivial solution.
For k = 3, by Theorem 3, ensuring condition c) for d 0,l (x, t) and W 0,l (x, t), we obtain problem By this we have determined the main term of the asymptotic. In addition, conditions b) of Theorem 3 gives From here we define v 1i (x, t) = 0, below it will be shown that P 1 i (x) = 0, therefore c 1 ii (x, t) = 0, and from the last equation we find c 1 ij (x, t) = 0, i = j. In the next k = 4 step free member F 4 (M ) = −T 1 u 2 + L ξ u 1 (M ) − ∂ t ∂ t u 0 . Satisfying condition c) of Theorem 3, we obtain the problems i.e. d 1,l i (x, t) = 0, W 1 ij , l(x, t) = 0, and, therefore, u 1 (M ) = 0. Regarding Y 3 i (N l ), W 3 ij (N l ) we obtain homogeneous equations, therefore We calculate and ensuring the solvability in U of the iteration equation for k = 4, we assume The solvability of the second system is ensured by the choice of P 0 (x): , as well as Under such assumptions, the second system of (23) is solvable. It is solved under the initial conditions (24) and the initial condition c 2 ii (x, t)| t=0 = −v 2i (x, 0) − P 2 i (x), which is obtained from (17). This completely defines the main term of the asymptotic.

ESTIMATES OF THE REMAINDER TERM
From the construction of solutions to iterative problems, it can be seen that the function g εn (x, t, ε) is uniformly bounded in Ω. Applying the maximum principle [28], we can establish a uniform estimate in Ω ||R εn (x, t, ε)|| < cε n+1 .