Solution of the nonregular problem for a parabolic equation with the time derivative in the boundary condition

Abstract

There is studied the Hölder space solution $u_{ε}$ of the problem for parabolic equation with the time derivative $ε \partial_{t} u_{ε} |_{Σ}$ in the boundary condition, where $ε > 0$ is a small parameter. The unique solvability of the perturbed problem and estimates of it’s solution are obtained. The convergence of $u_{ε}$ as $ε \to 0$ to the solution of the unperturbed problem is proved. Boundary layer is not appeared.

Keywords

Parabolic equation boundary–value problem small parameter in the boundary condition Hölder space existence uniqueness coercive estimates convergence of the solution

1. Statement of the problem

1.1. Statement of the problem

Let Ω be domain in $R^{n}$ , $n ⩾ 2$ , with the boundary Σ, $Ω_{T} : = Ω \times (0, T)$ , $Σ_{T} : = Σ \times [0, T]$ .

Let $A (x, t, \partial_{x})$ be the second order elliptic operator $\begin{matrix} A (x, t, \partial_{x}) : = \sum_{i, j = 1}^{n} a_{i j} (x, t) \partial_{x_{i} x_{j}}^{2} + \sum_{i = 1}^{n} a_{i} (x, t) \partial_{x_{i}} + a (x, t), \end{matrix}$ where $\begin{matrix} (1.1) & a_{i j} (x, t) = a_{j i} (x, t), i, j = 1, \dots, n, \sum_{i, j = 1}^{n} a_{i j} (x, t) ξ_{i} ξ_{j} ⩾ a_{0} ξ^{2} \end{matrix}$ $\forall ξ \in R^{n}$ , $\forall (x, t) \in {\overline{Ω}}_{T}$ , $a_{0} = const > 0$ .

We shall study the boundary – value problem with an unknown function $u (x, t)$ $\begin{array}{l} (1.2) & \partial_{t} u - A (x, t, \partial_{x}) u = f (x, t) in Ω_{T}, \\ (1.3) & u |_{t = 0} = u_{0} (x) in Ω, \\ (1.4) & (ε \partial_{t} u - \overline{b} (x, t) \nabla^{T} u + b_{0} (x, t) u) |_{Σ} = φ_{1} (x, t) + ε φ_{2} (x, t), t \in (0, T), \end{array}$ where $ε > 0$ is a small parameter, $\overline{b} (x, t) = (b_{1} (x, t), \dots, b_{n} (x, t))$ , $\nabla^{T} : = colon (\partial_{x_{1}}, \dots, \partial_{x_{n}})$ is a column – vector, $\overline{b} \nabla^{T} u = b_{1} \partial_{x_{1}} u + \dots + b_{n} \partial_{x_{n}} u$ is a scalar product, $\partial_{t} = \partial / \partial t$ , $\partial_{x_{i}} = \partial / \partial x_{i}$ .

In the boundary condition (1.4) there is term $ε φ_{2} (x, t)$ , justification of it is given in the Section 1.2.

For $ε = 0$ we have an unperturbed problem with an unknown function $w (x, t)$ $\begin{array}{l} (1.5) & \partial_{t} w - A (x, t, \partial_{x}) w = f (x, t) in Ω_{T}, \\ (1.6) & w |_{t = 0} = u_{0} (x) in Ω, \\ (1.7) & (- \overline{b} (x, t) \nabla^{T} w + b_{0} (x, t) w) |_{Σ} = φ_{1} (x, t), t \in (0, T) . \end{array}$

In these two problems vector $\overline{b} (x, t)$ satisfies the condition $\begin{matrix} (1.8) & \overline{b} (x, t) {\overline{ν}}^{T} (x) ⩾ d_{0} = const > 0 \forall (x, t) \in Σ_{T}, \end{matrix}$ where $\overline{ν} = (ν_{1} (x, t), \dots, ν_{n} (x, t))$ is a unit normal to the surface Σ directed into Ω.

We point out that (1.8) is the condition of the solvability of the problems (1.2)–(1.4) and (1.5)–(1.7).

The solution of the problem (1.2)–(1.4) depends on the small parameter $ε : u (x, t) = u_{ε} (x, t)$ . For the simplicity we omit index ε until it will be necessary.

The positive constants in every Section are designated by $C_{1}, C_{2}, \dots$ .

The problem (1.2)–(1.4) describes physical processes, such as heat and mass transfer [12], heat process with a thin film of ε width on the boundary Σ of a domain. Moreover, it has a wide application in the theory of free boundary problems, this problem is a linearized multidimensional one – phase free boundary problem of the Stefan type [1, 20, 26, 29]. The problem (1.2)–(1.4) is not embedded into the general theory of the initial boundary–value problems for the parabolic equations, because the Lopatinskiy condition is not fulfilled for it. If we assume the condition $\overline{b} (x, t) {\overline{ν}}^{T} (x) ⩽ - d_{0}$ , $d_{0} > 0$ , instead of (1.8), then the problem (1.2)–(1.4) becomes incorrect by Hadamard [32].

The problem (1.2)–(1.4) with $ε = 1$ was studied in the Sobolev and Hölder spaces by V.V. Barkovskiyv, V.L. Kulchitskiy, G.I. Bizhanova and V.A. Solonnikov, J. Kacur, M.I. Khazan, I.Sh. Mogilevskiy and N.A. Grigor’eva, B.M. Anisyutin, B.V. Bazaliy, E.V. Radkevich. The full bibliography is given in [8].

Perturbed problems have wide practical applications, they are the mathematical models of the real physical processes, and it is important to know how the process will go, when the small parameters (coefficients of the problems) vanish. From the other hand, such problems are rich in the mathematical sense.

The problems with the small parameters were and are studied by V.F. Butuzov, M. Chipot, F.J. Correa, A.R. Danilin, G.–M. Gie, S. Guesmia, M. Hamouda, M.I. Imanaliev, Ch.–Y. Jung, K.A. Kasymov, A.V. Nesterov, O.A. Oleinik, E. Park, A. Sengouga, R. Temam, A.N. Tikhonov, A.B. Vasil’eva, M.I. Vishik and L.A. Lyusternik, X. Wang (see [9–11, 13–19, 21–24, 27, 28, 31, 33–35]) and by many other mathematicians.

G.I. Bizhanova considered the model problems with the constant coefficients for parabolic equations with the small parameters in [3, 5, 6]. In these problems the given functions and solutions are subjected to the zero initial data. In particular, in [3] there was studied the problem in the half space $x_{n} > 0$ with the small parameter at the time derivative in the boundary condition. The obtained results are applied in the Appendix. The problems in the whole space $R^{n}$ with two small parameters in the conjugation conditions on the hyperplane $x_{n} = 0$ were considered in [5, 6]. The solutions of these problems were constructed in the explicit form with the help of the Laplace and Fourier transforms and by direct evaluation of these solutions Hölder space estimates of them were obtained. In [5] there was also proved the convergence of the solution as small parameters tend to zero.

We remark that the convergence of the solution of the problem (1.2)–(1.4) as $ε \to 0$ to the solution of the problem (1.5)–(1.7) is not obverse fact. Really, according to the Theorem 3.1 all the terms in the condition (1.4) are bounded functions, so $\partial_{t} u |_{Σ} = O (1 / ε)$ as $ε \to 0$ and the limit of $ε \partial_{t} u |_{Σ}$ as $ε \to 0$ is not known.

In Section 2 there are given the definitions of the Hölder spaces, deduced the compatibility conditions of the problem (1.2)–(1.4). The main results are formulated in Section 3. The problem (1.2)–(1.4) is reduced to the equivalent one with the given and unknown functions satisfying zero initial conditions in Sections 4. In Section 5 the unique solvability of the problem (1.2)–(1.4) and estimates of it’s solution are obtained. The convergence of the solution of the problem (1.2)–(1.4) as $ε \to 0$ to the solution of the problem (1.5)–(1.7) is proved in Section 6. In Appendix A there is considered the model problem, the obtained results are applied in the proofs of the main theorems.

1.2. Justification of the statement of the problem (1.2)–(1.4) with a small parameter in the right – hand side of the boundary condition

We have written the sum of the functions $φ_{1} (x, t) + ε φ_{2} (x, t)$ in (1.4) instead of $φ_{1} (x, t)$ (as it is usually considered in the problems with a small parameter) because of the compatibility conditions. This conditions guarantee the continuity of the solution and all admissible derivatives of it in the closure of the domain.The compatibility condition of the zero order for the problem (1.2)–(1.4) is as follows: $\begin{matrix} (ε (A (x, 0, \partial_{x}) u_{0} (x) + f (x, 0)) - \overline{b} (x, 0) \nabla^{T} u_{0} (x) + b_{0} (x, 0) u_{0} (x)) |_{Σ} \\ = φ_{1} (x, 0) + ε φ_{2} (x, 0) . \end{matrix}$

We see that without the term $ε φ_{2} (x, 0)$ this identity can not be fulfilled for $\forall ε \in (0, ε_{0}]$ for the fixed given functions.

In [7] G.I.Bizhanova and M.N.Shaimardanova studied one–dimensional model problem $\begin{array}{l} \partial_{t} V_{0} - a \partial_{x_{1}}^{2} V_{0} = 0, x_{1} > 0, 0 < t < T, \\ V_{0} |_{t = 0} = 0, x_{1} > 0, (\partial_{t} V_{0} - b_{1} \partial_{x_{1}} V_{0}) |_{x_{1} = 0} = A_{0}, 0 < t < T, \end{array}$ where $a > 0$ , $b_{1} > 0, A_{0}$ are constants. We see that in this problem there is not fulfilled the compatibility condition of the zero order: $0 \neq A_{0}$ . The derivatives $\begin{matrix} \partial_{t} V_{0} = a \partial_{x_{1}}^{2} V_{0} = A_{0} erfs \frac{x_{1}}{2 \sqrt{a t}} - A_{0} \frac{b_{1}}{\sqrt{a π}} J (x_{1}, t) \end{matrix}$ are bounded, but discontinuous functions at the point $x_{1} = 0$ , $t = 0$ , here $\begin{matrix} erfs z = \frac{2}{\sqrt{π}} \int_{z}^{\infty} e^{- ζ^{2}} d ζ, J (x_{1}, t) = \int_{0}^{\infty} \frac{1}{\sqrt{t - σ}} e^{- \frac{{(x_{1} + b_{1} σ)}^{2}}{4 a (t - σ)}} d σ . \end{matrix}$ The higher derivatives of the solution are singular functions, for instance, $\begin{array}{l} \partial_{t} \partial_{x_{1}} V_{0} (x_{1}, t) = a \partial_{x_{1}}^{3} V_{0} (x_{1}, t) = A_{0} (- 2 Γ (x_{1}, t) + \frac{b_{1}}{a} erfc \frac{x_{1}}{2 \sqrt{a t}} - \frac{b_{1}}{a} J (x_{1}, t)), \end{array}$ where $Γ (x_{1}, t) = \frac{1}{2 \sqrt{a t}} e^{- \frac{x_{1}^{2}}{4 a t}}$ is a fundamental solution of the heat equation $\partial_{t} V_{0} - a \partial_{x_{1}}^{2} V_{0} = 0$ .

In [4] there was studied one–dimensional problem for parabolic equation with variable coefficients and with a time derivative in the boundary condition, when the compatibility conditions of the zero and the first orders of initial and boundary data are not fulfilled. In [4, 7] there was proved that each of the solutions of the problems is represented as a sum of smooth and singular functions.

2. Definitions of the Hölder spaces, compatibility conditions of the problem (1.2)–(1.4)

2.1. Definitions of the Hölder spaces

We shall study the problem (1.2)–(1.4) in the Hölder space $C_{x t}^{l, l / 2} ({\overline{Ω}}_{T})$ , l is a positive non-integer, of the functions $u (x, t)$ with the norm [25] $\begin{array}{l} | u |_{Ω_{T}}^{(l)} : = & \sum_{2 m_{0} + | m | ⩽ [l]} {| \partial_{t}^{m_{0}} \partial_{x}^{m} u |}_{Ω_{T}} + \sum_{2 m_{0} + | m | = [l]} ({[\partial_{t}^{m_{0}} \partial_{x}^{m} u]}_{x, Ω_{T}}^{(α)} + {[\partial_{t}^{m_{0}} \partial_{x}^{m} u]}_{t, Ω_{T}}^{(α / 2)}) \\ (2.1) & + \{\begin{matrix} \sum_{2 m_{0} + | m | = [l] - 1} {[\partial_{t}^{m_{0}} \partial_{x}^{m} u]}_{t, Ω_{T}}^{(\frac{1 + α}{2})}, & [l] ⩾ 1, \\ 0, & [l] = 0, \end{matrix} \end{array}$ where $l - [l] = α \in (0, 1)$ , $m = (m_{1}, \dots, m_{n})$ , $m_{1}, \dots, m_{n}$ are non-negative integers, $| m | = m_{1} + \dots + m_{n}$ , $\begin{array}{l} | v |_{Ω_{T}} = max_{(x, t) \in \overline{Ω}} | v |, {[v]}_{x, Ω_{T}}^{(α)} = max_{(x, t), (z, t) \in {\overline{Ω}}_{T}} \frac{| v (x, t) - v (z, t) |}{| x - z |^{α}}, \\ {[v]}_{t, Ω_{T}}^{(α)} = max_{(x, t), (x, t_{1}) \in {\overline{Ω}}_{T}} \frac{| v (x, t) - v (x, t_{1}) |}{| t - t_{1} |^{α}} . \end{array}$

By ${\overset{\circ}{C}}_{x t}^{l, l / 2} ({\overline{Ω}}_{T})$ we designate the subset of the functions $u (x, t) \in C_{x t}^{l, l / 2} ({\overline{Ω}}_{T})$ , such that $\partial_{t}^{m_{0}} u |_{t = 0} = 0$ , $m_{0} = 0, \dots, [l / 2]$ .

We shall make use of the following lemma in the Appendix.

Lemma 2.1 ([30]).

In ${\overset{\circ}{C}}_{x t}^{k + α, \frac{k + α}{2}} ({\overline{Ω}}_{T})$ , $k = 0, 1, \dots$ , $α \in (0, 1)$ , the norm $| u |_{Ω_{T}}^{(k + α)}$ defined by the formula ( 2.1 ) is equivalent to the norm $\begin{array}{l} ‖ u ‖_{Ω_{T}}^{(k + α)} = & sup_{(x, t) \in Ω_{T}} t^{- \frac{k + α}{2}} | u | + \sum_{2 m_{0} + | m | = k} ({[\partial_{t}^{m_{0}} \partial_{x}^{m} u]}_{x, Ω_{T}}^{(α)} + {[\partial_{t}^{m_{0}} \partial_{x}^{m} u]}_{t, Ω_{T}}^{(α / 2)}) \\ + \{\begin{matrix} \sum_{2 m_{0} + | m | = k - 1} {[\partial_{t}^{m_{0}} \partial_{x}^{m} u]}_{t, Ω_{T}}^{(\frac{1 + α}{2})}, & k ⩾ 1, \\ 0, & k = 0 . \end{matrix} \end{array}$

2.2. Compatibility conditions of the problem (1.2)–(1.4)

To solve the problems in the Hölder spaces it is necessary that the compatibility conditions of the initial and boundary data are fulfilled. We define the compatibility conditions for the problem (1.2)–(1.4).

We find the time derivatives $\partial_{t}^{p} u (x, t) |_{t = 0}, p = 1, 2, \dots$ , from an equation (1.2) with the help of the initial data (1.3) $\begin{array}{l} (2.2) & \begin{array}{l} u (x, t) |_{t = 0} = u_{0} (x), \\ \partial_{t} u (x, t) |_{t = 0} : = F_{1}^{*} (x, t) |_{t = 0} = A (x, 0, \partial_{x}) u_{0} (x) + f (x, 0), \\ \begin{matrix} \partial_{t}^{2} u (x, t) |_{t = 0} : = & F_{2}^{*} (x, t) |_{t = 0} \\ = & \partial_{t} A (x, t, \partial_{x}) |_{t = 0} u_{0} (x) \\ + A (x, 0, \partial_{x}) (A (x, 0, \partial_{x}) u_{0} (x) + f (x, 0)) + f_{t} (x, t) |_{t = 0}, \end{matrix} \\ \begin{matrix} \partial_{t}^{1 + p} u (x, t) |_{t = 0} : = & F_{1 + p}^{*} (x, t) |_{t = 0} \\ = & (\partial_{t}^{p} (A (x, t, \partial_{x}) u) + \partial_{t}^{p} f (x, t)) |_{t = 0} \\ \equiv & \sum_{q = 0}^{p} C_{p}^{q} (\partial_{t}^{p - q} A (x, t, \partial_{x})) \partial_{t}^{q} u (x, t) |_{t = 0} + \partial_{t}^{p} f (x, t) |_{t = 0} \\ = & \partial_{t}^{p} A (x, t, \partial_{x}) |_{t = 0} u_{0} (x)) + \sum_{q = 1}^{p} C_{p}^{q} \partial_{t}^{p - q} A (x, t, \partial_{x}) |_{t = 0} F_{q}^{*} (x, t) |_{t = 0} \\ + \partial_{t}^{p} f (x, t) |_{t = 0}, \end{matrix} \end{array} \end{array}$ where $C_{p}^{q} = \frac{p!}{q! (p - q)!}$ are the binomial coefficients. We see that the derivatives $\partial_{t}^{1 + p} u (x, t) |_{t = 0}$ are expressed via the derivatives of the functions $u_{0} (x)$ and $f (x, t)$ at $t = 0$ .

Let $\begin{matrix} (2.3) & F_{1 + p} (x) : = F_{1 + p}^{*} (x, t) |_{Σ, t = 0} \equiv \partial_{t}^{1 + p} u (x, t) |_{Σ, t = 0} . \end{matrix}$

Consider boundary condition (1.4). Differentiating it with respect to t and taking into consideration the formula (2.2) we shall have $\begin{array}{l} (ε \partial_{t}^{1 + p} u (x, t) - \partial_{t}^{p} (\overline{b} (x, t) \nabla^{T} u (x, t)) + \partial_{t}^{p} (b_{0} (x, t) u (x, t))) |_{Σ, t = 0} \\ \equiv (ε \partial_{t}^{1 + p} u (x, t) - \sum_{q = 0}^{p} C_{p}^{q} (\partial_{t}^{p - q} \overline{b} (x, t)) \nabla^{T} \partial_{t}^{q} u (x, t) \\ + \sum_{q = 0}^{p} C_{p}^{q} (\partial_{t}^{p - q} b_{0} (x, t)) \partial_{t}^{p - q} u (x, t)) |_{Σ, t = 0} \\ (2.4) & = \partial_{t}^{p} φ_{1} (x, t) |_{t = 0} + ε \partial_{t}^{p} φ_{2} (x, t) |_{t = 0} . \end{array}$

We can see that an expression (2.4) without the term $ε \partial_{t}^{p} φ_{2} (x, t) |_{t = 0}$ has not sense, because ε is a small parameter, it moves in $[0, ε_{0}]$ , and given functions have to be fixed.

We shall say that in the problem (1.2)–(1.4) there is fulfilled the compatibility condition of the initial and boundary data of p–order for all $ε \in (0, ε_{0}]$ , $p = 0, 1, \dots$ , if the following identity takes place: $\begin{array}{l} (- \partial_{t}^{p} (\overline{b} (x, t) \nabla^{T} u (x, t)) + \partial_{t}^{p} (b_{0} (x, t) u (x, t))) |_{Σ, t = 0} \\ \equiv (- \sum_{q = 0}^{p} C_{p}^{q} (\partial_{t}^{p - q} \overline{b} (x, t)) \nabla^{T} \partial_{t}^{q} u (x, t) + \sum_{q = 0}^{p} C_{p}^{q} (\partial_{t}^{p - q} b_{0} (x, t)) \partial_{t}^{q} u (x, t)) |_{Σ, t = 0} \\ (2.5) & = \partial_{t}^{p} φ_{1} (x, t) |_{t = 0}, \\ (2.6) & ε \partial_{t}^{1 + p} u (x, t) |_{Σ, t = 0} = ε \partial_{t}^{p} φ_{2} (x, t) |_{t = 0} . \end{array}$

We rewrite (2.5), (2.6) taking into account formula (2.3) $\begin{array}{l} (2.7) & \begin{matrix} - (\partial_{t}^{p} \overline{b} (x, t) \nabla^{T} u_{0} (x) - \partial_{t}^{p} b_{0} (x, t) u_{0} (x)) |_{Σ, t = 0} \\ - (\sum_{q = 1}^{p} C_{p}^{q} (\partial_{t}^{p - q} \overline{b} (x, t)) |_{Σ, t = 0} \nabla^{T} F_{q} (x) - \sum_{q = 1}^{p} C_{p}^{q} (\partial_{t}^{p - q} b_{0} (x, t)) |_{Σ, t = 0} F_{q} (x)) \\ = \partial_{t}^{p} φ_{1} (x, t) |_{t = 0}, \end{matrix} \\ (2.8) & ε F_{1 + p} (x) = ε \partial_{t}^{p} φ_{2} (x, t) |_{t = 0}, x \in Σ . \end{array}$

In particular, the compatibility conditions of the zero and the first orders are $\begin{matrix} (2.9) & \begin{matrix} (- \overline{b} (x, 0) \nabla^{T} u_{0} (x) + b_{0} (x, 0) u_{0} (x)) |_{Σ} = φ_{1} (x, 0), \\ ε (A (x, 0, \partial_{x}) u_{0} (x) + f (x, 0)) |_{Σ} = ε φ_{2} (x, 0) \end{matrix} \end{matrix}$ and $\begin{array}{l} (2.10) & \begin{array}{l} - (\partial_{t} \overline{b} (x, t) \nabla^{T} u_{0} (x) - \partial_{t} b_{0} (x, t)) u_{0} (x)) |_{\begin{array}{c} Σ, \\ t = 0 \end{array}} \\ - (\overline{b} (x, 0) \nabla^{T} u_{0} (x) - b_{0} (x, 0)) (A (x, 0, \partial_{x}) u_{0} (x) + f (x, 0)) |_{Σ} = \partial_{t} φ_{1} (x, t) |_{t = 0}, \\ ε (A_{t} (x, t, \partial_{x}) u_{0} (x) + A (x, 0, \partial_{x}) (A (x, 0, \partial_{x}) u_{0} (x) + f (x, 0)) + \partial_{t} f (x, t)) |_{x \in Σ, t = 0} \\ = ε \partial_{t} φ_{2} (x, t) |_{t = 0} . \end{array} \end{array}$

The equality (2.5) is the compatibility condition of p – order of the unperturbed problem (1.5)–(1.7).

The identity (2.4) with $ε = 1$ is the compatibility condition of p–order of the problem (1.2)–(1.4) with $ε = 1$ $\begin{array}{l} (F_{1 + p} (x) - \partial_{t}^{p} (\overline{b} (x, t) \nabla^{T} u (x, t) - b_{0} (x, t) u (x, t))) |_{x \in Σ, t = 0} \\ (2.11) & = \partial_{t}^{p} (φ_{1} (x, t) + φ_{2} (x, t)) |_{t = 0}, p = 0, 1, \dots . \end{array}$

3. Main results

Let $\begin{array}{l} (3.1) & \begin{aligned} a_{i j} (x, t), a_{i} (x, t), a (x, t) \in C_{x t}^{l, l / 2} ({\overline{Ω}}_{T}), i, j = 1, \dots, n, \\ b_{q} (x, t) \in C_{x t}^{1 + l, \frac{1 + l}{2}} (Σ_{T}), q = 0, 1, \dots, n, \end{aligned} \\ (3.2) & ε F_{1 + p} (x) \in C^{1 + l - 2 p} (Σ), p = 0, \dots, [\frac{1 + l}{2}], \end{array}$ where the functions $F_{1 + p} (x) = : \partial_{t}^{1 + p} u |_{Σ, t = 0}$ are determined by the formulas (2.3), (2.2) and depend on the functions $u_{0}$ and f and their derivatives.

Now we formulate the main results.

Theorem 3.1.
Let $Σ \in C^{2 + l}$ , l be positive non-integer, $0 < ε ⩽ ε_{0}$ , the conditions ( 1.8 ), ( 3.1 ), ( 3.2 ) be fulfilled.

For any functions $f (x, t) \in C_{x t}^{l, l / 2} ({\overline{Ω}}_{T})$ , $u_{0} (x) \in C^{2 + l} (\overline{Ω})$ , $φ_{j} (x, t) \in C_{x t}^{1 + l, \frac{1 + l}{2}} (Σ_{T})$ , $j = 1, 2$ , satisfying the compatibility conditions ( 2.7 ), ( 2.8 ) of the $0, \dots, [\frac{1 + l}{2}]$ – orders the problem ( 1.2 )–( 1.4 ) has a unique solution $u (x, t) \in C_{x t}^{2 + l, 1 + l / 2} ({\overline{Ω}}_{T})$ , $ε \partial_{t} u (x, t) \in C_{x t}^{1 + l, \frac{1 + l}{2}} (Σ_{T})$ , and an estimate holds $\begin{array}{l} | u |_{Ω_{T}}^{(2 + l)} + | ε \partial_{t} u |_{Σ_{T}}^{(1 + l)} ⩽ & C_{2} (| u_{0} |_{Ω}^{(2 + l)} + | f |_{Ω_{T}}^{(l)} + | φ_{1} |_{Σ_{T}}^{(1 + l)}) \\ (3.3) & + C_{3} ε (| φ_{2} |_{Σ_{T}}^{(1 + l)} + \sum_{p = 0}^{[\frac{1 + l}{2}]} | F_{1 + p} |_{Σ}^{(1 + l - 2 p)}), \end{array}$ where the constants $C_{2}$ , $C_{3}$ do not depend on ε.

Formula (3.3) gives the asymptotic behavior with respect to ε of the solution $u (x, t)$ .

The conditions (3.2) correspond to the smoothness of the time derivatives of $ε \partial_{t} u (x, t) |_{Σ}$ . Conditions (3.2) follow also from the compatibility conditions (2.6): $ε F_{1 + p} (x) = ε \partial_{t}^{p} φ_{2} (x, t) |_{t = 0}$ with $φ_{2} (x, t) \in C_{x t}^{1 + l, \frac{1 + l}{2}} (Σ_{T})$ .

We point out that the requirement (3.2) can be fulfilled. For example, consider $F_{1} (x) = (A (x, 0, \partial_{x}) u_{0} (x) + f (x, 0)) |_{Σ}$ . Let $f (x, 0) = - A (x, 0, \partial_{x}) u_{0} (x) + f_{1} (x, 0)$ , here $f (x, 0), A (x, 0, \partial_{x}) u_{0} (x) \in C^{l} (\overline{Ω})$ . We suppose $f_{1} (x, 0) \in C^{1 + l} (\overline{Ω})$ , then we shall have $F_{1} (x) = f_{1} (x, 0) |_{Σ}$ .
Theorem 3.2.
Let $ε = 1$ , $Σ \in C^{2 + l}$ , l be positive non-integer, the conditions ( 1.8 ), ( 3.1 ), ( 3.2 ) with $ε = 1$ be fulfilled.

For any functions $f (x, t) \in C_{x t}^{l, l / 2} ({\overline{Ω}}_{T})$ , $u_{0} (x) \in C^{2 + l} (\overline{Ω})$ , $φ_{j} (x, t) \in C_{x t}^{1 + l, \frac{1 + l}{2}} (Σ_{T})$ , $j = 1, 2$ , satisfying the compatibility conditions ( 2.11 ) of the $0, \dots, [\frac{1 + l}{2}]$ – orders the problem ( 1.2 )–( 1.4 ) has a unique solution $u (x, t) \in C_{x t}^{2 + l, 1 + l / 2} ({\overline{Ω}}_{T})$ , $\partial_{t} u (x, t) \in C_{x t}^{1 + l, \frac{1 + l}{2}} (Σ_{T})$ , and an estimate holds $\begin{matrix} | u |_{Ω_{T}}^{(2 + l)} + | \partial_{t} u |_{Σ_{T}}^{(1 + l)} ⩽ C_{4} (| u_{0} |_{Ω}^{(2 + l)} + | f |_{Ω_{T}}^{(l)} + \sum_{j = 1}^{2} | φ_{j} |_{Σ_{T}}^{(1 + l)} + \sum_{p = 0}^{[\frac{1 + l}{2}]} | F_{1 + p} |_{Σ}^{(1 + l - 2 p)}) . \end{matrix}$

This theorem was proved in [8], here it follows from the Theorem 3.1.
Theorem 3.3.
Let $0 < ε ⩽ ε_{0}$ , $k = 0, 1, \dots$ , $α \in (0, 1)$ , let the conditions of the Theorem 3.1 be fulfilled with $l = k + α$ .

Then the time derivative $ε \partial_{t} u |_{Σ_{T}}$ in the boundary condition of the problem ( 1.2 )–( 1.4 ) satisfies an estimate for $\forall β \in (0, α / 2)$ $\begin{array}{l} | ε \partial_{t} u |_{Σ_{T}}^{(1 + k + β)} ⩽ & C_{5} ε^{\frac{α - β}{2}} (| u_{0} |_{Ω}^{(2 + k + α)} + | f |_{Ω_{T}}^{(k + α)} + | φ_{1} |_{Σ_{T}}^{(1 + k + α)}) \\ (3.4) & + C_{6} ε (| φ_{2} |_{Σ_{T}}^{(1 + k + α)} + \sum_{p = 0}^{[\frac{1 + k}{2}]} | F_{1 + p} |_{Σ}^{(1 + k + α - 2 p)}), \end{array}$ where the constants $C_{5}$ , $C_{6}$ do not depend on ε.

From this theorem we have $\begin{matrix} \partial_{t} u (x, t) |_{Σ} = O (ε^{- 1 + \frac{α - β}{2}}) as ε \to 0, \end{matrix}$ that is the time derivative $\partial_{t} u |_{Σ}$ is singular function with respect to ε.

We have obtained $ε^{\frac{α - β}{2}}, β \in (0, α / 2)$ , in the right– hand side of the inequality (3.4), because we estimate the norm of $ε \partial_{t} u |_{Σ}$ in the space $C_{x t}^{1 + k + β, \frac{1 + k + β}{2}} (Σ_{T})$ instead of $C_{x t}^{1 + k + α, \frac{1 + k + α}{2}} (Σ_{T})$ as in Theorem 3.1.

Theorems 3.1, 3.3 will be proved in Section 5.
Theorem 3.4.
Let all the conditions of Theorem 3.1 be fulfilled.

Then the solution $u_{ε} (x, t)$ of the problem ( 1.2 )–( 1.4 ) converges as $ε \to 0$ to the unique solution $w (x, t)$ of the problem ( 1.5 )–( 1.7 ), function $w (x, t)$ belongs to the space $C_{x t}^{2 + l, 1 + l / 2} ({\overline{Ω}}_{T})$ and satisfies an estimate $\begin{matrix} (3.5) & | w |_{Ω_{T}}^{(2 + l)} ⩽ C_{7} (| f |_{Ω_{T}}^{(l)} + | u_{0} |_{Ω}^{(2 + l)} + | φ_{1} |_{Σ_{T}}^{(1 + l)}) . \end{matrix}$
Consequence 3.4.1.
Let $Σ \in C^{2 + l}$ , l be positive non-integer, the conditions ( 1.8 ), ( 3.1 ) be fulfilled.

For any functions $u_{0} (x) \in C^{2 + l} (\overline{Ω})$ , $f (x, t) \in C_{x t}^{l, l / 2} ({\overline{Ω}}_{T})$ , $φ_{1} (x, t) \in C_{x t}^{1 + l, \frac{1 + l}{2}} (Σ_{T})$ satisfying the compatibility conditions ( 2.5 ) of the $0, \dots, [\frac{1 + l}{2}]$ – orders the problem ( 1.5 )–( 1.7 ) has unique solution $w (x, t) \in C_{x t}^{2 + l, 1 + l / 2} ({\overline{Ω}}_{T})$ , and an estimate ( 3.5 ) for it holds.

Theorem 3.4 is proved with the help of the Theorems 3.1, 3.3, and the limit function $w (x, t)$ is of the maximal regularity.

In the next theorem we shall obtain the solution $w (x, t)$ of the unperturbed problem with the help of the Theorem 3.1 and requirement of the higher smoothness of the given functions, than in the Theorem 3.4, and we lose the smoothness of the given functions.

Let the coefficients of the equation (1.2) and boundary condition (1.4) satisfy the following assumptions: $\begin{matrix} (3.6) & \begin{matrix} a_{i j} (x, t), a_{i} (x, t), a (x, t) \in C_{x t}^{1 + k + α, \frac{1 + k + α}{2}} ({\overline{Ω}}_{T}), i, j = 1, \dots, n, k = 0, 1, \dots, \\ b_{q} (x, t) \in C_{x t}^{2 + k + α, 1 + \frac{k + α}{2}} (Σ_{T}), q = 0, 1, \dots, n, \\ ε F_{1 + p} (x) \in C^{2 + k + α - 2 p} (Σ), p = 0, \dots, 1 + [k / 2], α \in (0, 1) . \end{matrix} \end{matrix}$
Theorem 3.5.
Let $Σ \in C^{3 + k + α}$ , $0 < ε ⩽ ε_{0}$ , the conditions ( 1.8 ), ( 3.6 ) be fulfilled.

Let $f (x, t) \in C_{x t}^{1 + k + α, \frac{1 + k + α}{2}} ({\overline{Ω}}_{T})$ , $u_{0} (x) \in C^{3 + k + α} (\overline{Ω})$ , $φ_{j} (x, t) \in C_{x t}^{2 + k + α, 1 + \frac{k + α}{2}} (Σ_{T})$ , $j = 1, 2$ , and the compatibility conditions ( 2.5 ), ( 2.6 ) of the $0, \dots, 1 + [k / 2]$ – orders take place.

Then the solution $u_{ε} (x, t)$ of the problem ( 1.2 )–( 1.4 ) converges as $ε \to 0$ to the unique solution $w (x, t)$ of the problem ( 1.5 )–( 1.7 ), the function $w (x, t)$ belongs to the space $C_{x t}^{2 + k + γ, 1 + k / 2 + γ / 2} ({\overline{Ω}}_{T}), γ = 1$ (Lipschitz index), and an estimate for it holds $\begin{matrix} | w |_{Ω_{T}}^{(2 + k + γ)} ⩽ C_{10} (| f |_{Ω_{T}}^{(1 + k + α)} + | u_{0} |_{Ω}^{(3 + k + α)} + | φ_{1} |_{Σ_{T}}^{(2 + k + α)}) . \end{matrix}$

Theorem 3.5 is proved without applying of an additional estimate (3.4) of the derivative $ε u_{ε} (x, t) |_{Σ}$ .

We shall prove Theorems 3.4, 3.5 in Section 6.

From the Theorems 3.1, 3.3–3.5 it follows that boundary layer does not appear.
4. Reduction of the problem (1.2)–(1.4) to the equivalent one

With the help of construction of the auxiliary functions we reduce the problem (1.2)–(1.4) to the equivalent one with the given functions and solution satisfying zero initial conditions. This is possible due to the compatibility conditions and give the possibility to find the estimates of the solution.

Lemma 4.1.
Let the conditions of Theorem 3.1 be fulfilled.

The problem ( 1.2 )–( 1.4 ) is equivalent to the following one with an unknown function $v (x, t)$ $\begin{array}{l} (4.1) & \partial_{t} v - A (x, t, \partial_{x}) v = 0 in Ω_{T}, \\ (4.2) & v |_{t = 0} = 0 in Ω, \\ (4.3) & (ε \partial_{t} v - \overline{b} (x, t) \nabla^{T} v + b_{0} (x, t) v) |_{Σ} = Φ_{1} (x, t) + ε Φ_{2} (x, t), t \in (0, T), \end{array}$ where $\begin{array}{l} (4.4) & Φ_{1} (x, t) : = φ_{1} (x, t) + (\overline{b} (x, t) \nabla^{T} V (x, t) - b_{0} (x, t) V (x, t)) |_{Σ}, \\ (4.5) & Φ_{2} (x, t) : = φ_{2} (x, t) - \partial_{t} V |_{Σ}, \\ V (x, t) |_{t = 0} = u_{0} (x), \partial_{t}^{1 + p} V |_{Σ, t = 0} = F_{1 + p} (x), p = 0, \dots, [\frac{1 + l}{2}], \end{array}$ the functions $F_{1 + p} (x) = : \partial_{t}^{1 + p} u (x, t) |_{Σ, t = 0}$ are determined by the formulas ( 2.3 ), ( 2.2 ), $V (x, t) \in C_{x t}^{2 + l, 1 + l / 2} ({\overline{Ω}}_{T})$ , $\partial_{t} V (x, t) |_{Σ} \in C_{x t}^{1 + l, \frac{1 + l}{2}} (Σ_{T})$ , $Φ_{j} (x, t) \in {\overset{\circ}{C}}_{x t}^{1 + l, \frac{1 + l}{2}} (Σ_{T})$ , $j = 1, 2$ , and $\begin{array}{l} (4.6) & | Φ_{1} |_{Σ_{T}}^{(1 + l)} ⩽ C_{1} (| u_{0} |_{Ω}^{(2 + l)} + | f |_{Ω_{T}}^{(l)} + | φ_{1} |_{Σ_{T}}^{(1 + l)}), \\ (4.7) & ε | Φ_{2} |_{Σ_{T}}^{(1 + l)} ⩽ C_{2} ε (| φ_{2} |_{Σ_{T}}^{(1 + l)} + \sum_{p = 0}^{[\frac{1 + l}{2}]} | F_{1 + p} |_{Σ}^{(1 + l - 2 p)}), \\ (4.8) & | V |_{Ω_{T}}^{(2 + l)} ⩽ C_{3} (| u_{0} |_{Ω}^{(2 + l)} + | f |_{Ω_{T}}^{(l)}), | \partial_{t} V |_{Σ_{T}}^{(1 + l)} ⩽ C_{4} \sum_{p = 0}^{[\frac{1 + l}{2}]} | F_{p} |_{Σ}^{(1 + l - 2 p)}, \end{array}$ the constants $C_{1}$ – $C_{4}$ do not depend on ε.
Proof.
The functions $F_{1 + p} (x)$ are determined on the boundary Σ and belong to the space $C^{1 + l - 2 p} (Σ)$ , $p = 0, \dots, [\frac{1 + l}{2}]$ . We extend them as ${\tilde{F}}_{1 + p} (x)$ into $R^{n}$ with preserving of their smoothness and such that ${\tilde{F}}_{1 + p} (x) \in C^{1 + l - 2 p} (R^{n})$ , $\begin{matrix} (4.9) & {\tilde{F}}_{1 + p} (x) |_{Σ} = F_{1 + p} (x), | {\tilde{F}}_{1 + p} |_{R^{n}}^{(1 + l - 2 p)} ⩽ C_{5} | F_{1 + p} |_{Σ}^{(1 + l - 2 p)}, p = 0, \dots, [\frac{1 + l}{2}] . \end{matrix}$

Consider initial problem with unknown function ${\tilde{V}}_{0} (x, t)$ in $R_{T}^{n} = : R^{n} \times (0, T)$ $\begin{matrix} (4.10) & {\tilde{V}}_{0} |_{t = 0} = 0, \partial_{t}^{1 + p} {\tilde{V}}_{0} |_{t = 0} = {\tilde{F}}_{1 + p} (x), p = 0, \dots, [\frac{1 + l}{2}], x \in R^{n} . \end{matrix}$

By [25], Ch.IV, §4, Theorem 4.3, the problem (4.10) has unique solution ${\tilde{V}}_{0} (x, t) \in C_{x t}^{3 + l, \frac{3 + l}{2}} (R_{T}^{n})$ , and $\begin{matrix} (4.11) & | {\tilde{V}}_{0} |_{R_{T}^{n}}^{(3 + l)} ⩽ C_{6} \sum_{p = 0}^{[\frac{1 + l}{2}]} | {\tilde{F}}_{1 + p} |_{R^{n}}^{(1 + l - 2 p)} ⩽ C_{7} \sum_{p = 0}^{[\frac{1 + l}{2}]} | F_{1 + p} |_{Σ}^{(1 + l - 2 p)} . \end{matrix}$

Let $\begin{matrix} (4.12) & V_{0} (x, t) = : {\tilde{V}}_{0} (x, t) |_{x \in Σ} . \end{matrix}$

We have due to (4.9)–(4.12) $\begin{matrix} V_{0} |_{t = 0} = 0, \partial_{t}^{1 + p} V_{0} (x, t) |_{t = 0} \equiv \partial_{t}^{1 + p} {\tilde{V}}_{0} |_{x \in Σ, t = 0} = {\tilde{F}}_{1 + p} (x) |_{x \in Σ} = F_{1 + p} (x), \end{matrix}$ $p = 0, \dots, [\frac{1 + l}{2}]$ , $V_{0} (x, t) \in C_{x t}^{3 + l, \frac{3 + l}{2}} (R_{T}^{n})$ , and $\begin{matrix} (4.13) & | V_{0} |_{Σ_{T}}^{(3 + l)} = | {\tilde{V}}_{0} |_{Σ_{T}}^{(3 + l)} ⩽ C_{8} \sum_{p = 0}^{[\frac{1 + l}{2}]} | F_{1 + p} |_{Σ}^{(1 + l - 2 p)} . \end{matrix}$

We consider another auxiliary first boundary–value problem with an unknown function $V (x, t)$ $\begin{array}{l} (4.14) & \partial_{t} V - A (x, t, \partial_{x}) V = f (x, t) in Ω_{T}, \\ (4.15) & V |_{t = 0} = u_{0} (x) in Ω, \\ (4.16) & V |_{Σ} = V_{0} (x, t) + u_{0} (x) |_{Σ}, t \in (0, T) . \end{array}$

The compatibility conditions are fulfilled in the problem (4.14)–(4.16). Really, from the boundary condition (4.16) with the help of (4.15) and condition $V_{0} |_{t = 0} = 0$ we find $\begin{matrix} V |_{Σ, t = 0} \equiv u_{0} (x) |_{Σ} = u_{0} (x) |_{Σ}, \end{matrix}$ we differentiate the condition (4.16) with respect to t and apply (4.14), (4.15), then $\begin{array}{l} \partial_{t} V |_{Σ, t = 0} \equiv A (x, t, \partial_{x}) u_{0} (x) |_{Σ, t = 0} + f (x, 0) |_{Σ} \equiv F_{1} (x) = \partial_{t} V_{0} |_{t = 0} \equiv F_{1} (x), \\ \partial_{t}^{1 + p} V |_{Σ, t = 0} \equiv F_{1 + p} (x) = \partial_{t}^{1 + p} V_{0} |_{t = 0} \equiv F_{1 + p} (x) . \end{array}$

These identities are the compatibility conditions of $0, \dots, [\frac{1 + l}{2}]$ – orders.

The problem (4.14)–(4.16) has unique solution [25] $V (x, t) \in C_{x t}^{2 + l, 1 + l / 2} ({\overline{Ω}}_{T})$ and it satisfies an estimate $\begin{matrix} (4.17) & | V |_{Ω_{T}}^{(2 + l)} ⩽ C_{9} (| u_{0} |_{Ω}^{(2 + l)} + | f |_{Ω_{T}}^{(l)} + | V_{0} |_{Σ_{T}}^{(2 + l)}) . \end{matrix}$

We have an estimate (4.13) for the function $V_{0} (x, t)$ , with the help of it we obtain $\begin{matrix} (4.18) & | V_{0} |_{Σ_{T}}^{(2 + l)} ⩽ C_{10} \sum_{p = 0}^{[l / 2]} | F_{1 + p} |_{Σ}^{(l - 2 p)} ⩽ C_{11} (| u_{0} |_{Ω}^{(2 + l)} + | f |_{Ω_{T}}^{(l)}), \end{matrix}$ and from (4.17), (4.18) we find $\begin{matrix} (4.19) & | V |_{Ω_{T}}^{(2 + l)} ⩽ C_{12} (| u_{0} |_{Ω}^{(2 + l)} + | f |_{Ω_{T}}^{(l)}) . \end{matrix}$

We find $\partial_{t} V (x, t) |_{Σ} = \partial_{t} V_{0} (x, t) \in C_{x t}^{1 + l, \frac{1 + l}{2}} (Σ_{T})$ from the boundary condition (4.16) and then we shall have due to (4.13) $\begin{matrix} (4.20) & | \partial_{t} V |_{Σ_{T}}^{(1 + l)} = | \partial_{t} V_{0} |_{Σ_{T}}^{(1 + l)} ⩽ C_{13} \sum_{p = 0}^{[\frac{1 + l}{2}]} | F_{p} |_{Σ}^{(1 + l - 2 p)} . \end{matrix}$

(4.19), (4.20) give the estimates (4.8).

Now in the problem (1.2)–(1.4) we make the substitution $\begin{matrix} (4.21) & u (x, t) = v (x, t) + V (x, t), \end{matrix}$ then for the new unknown function v we obtain the problem (4.1)–(4.3).

It follows from the equation (4.1) and initial condition (4.2) that the solution of the problem (4.1)–(4.3) and all it’s admissible derivatives satisfy zero initial conditions.

Consider the functions $Φ_{1}$ , $Φ_{2}$ determined by the formulas (4.4), (4.5) We shall show that the functions $Φ_{1}$ , $Φ_{2}$ satisfy zero initial data. We take into consideration that $V (x, t)$ is the solution of the problem (4.14)–(4.16).

With the help of the compatibility conditions (2.9), (2.10), (2.7) we derive $\begin{array}{l} \begin{aligned} Φ_{1} |_{t = 0} & \equiv φ_{1} |_{t = 0} + (\overline{b} (x, t) \nabla^{T} V - b_{0} (x, t) V) |_{Σ, t = 0} \\ \equiv φ_{1} |_{t = 0} + (\overline{b} (x, 0) \nabla^{T} u_{0} (x) - b_{0} (x, t) u_{0}) |_{Σ} = 0, \end{aligned} \\ \partial_{t}^{p} Φ_{1} |_{t = 0} = \partial_{t}^{p} φ_{1} |_{t = 0} + (\partial_{t}^{p} (\overline{b} (x, t) V) - \partial_{t}^{p} (b_{0} (x, t) V) |_{Σ, t = 0} \end{array}$ and $\begin{array}{l} \partial_{t}^{p} Φ_{1} |_{t = 0} \equiv & \partial_{t}^{p} φ_{1} |_{t = 0} + (\partial_{t}^{p} \overline{b} (x, t) \nabla^{T} u_{0} (x) - \partial_{t}^{p} b_{0} (x, t) u_{0} (x)) |_{Σ, t = 0} \\ + \sum_{q = 1}^{p} C_{p}^{q} (\partial_{t}^{p - q} \overline{b} (x, t) \nabla^{T} F_{q} (x) - \partial_{t}^{p - q} b_{0} (x, t) F_{q} (x)) |_{Σ, t = 0} \\ = & 0, p = 1, \dots, [\frac{1 + l}{2}] . \end{array}$

We consider the function $Φ_{2} (x, t)$ $\begin{array}{l} Φ_{2} (x, t) |_{t = 0} \equiv φ_{2} |_{t = 0} - \partial_{t} V |_{Σ, t = 0} \equiv φ_{2} |_{t = 0} - F_{1} (x) = 0, \\ \partial_{t}^{p} Φ_{2} (x, t) |_{t = 0} \equiv \partial_{t}^{p} φ_{2} |_{t = 0} - \partial_{t}^{1 + p} V |_{Σ, t = 0} \equiv \partial_{t}^{p} φ_{2} |_{t = 0} - F_{1 + p} (x) = 0, \end{array}$ $p = 1, \dots, [\frac{1 + l}{2}]$ , here we have made use of the compatibility conditions (2.9), (2.10), (2.8): $ε \partial_{t}^{1 + p} u (x, t) |_{x \in Σ, t = 0} \equiv ε F_{1 + p} (x) = ε \partial_{t}^{p} φ_{2} (x, t) |_{t = 0}$ and condition $\partial_{t}^{1 + p} V (x, t) |_{x \in Σ, t = 0} = F_{1 + p} (x)$ .

Thus, we have proved $\begin{matrix} \partial_{t}^{p} Φ_{j} (x, t) |_{t = 0} = 0, j = 1, 2, p = 0, 1, \dots, [\frac{1 + l}{2}], \end{matrix}$ that is the functions $Φ_{1}$ , $Φ_{2}$ satisfy zero initial data.

We can see that the functions $Φ_{1}$ , $Φ_{2}$ belong to the space ${\overset{\circ}{C}}_{x t}^{1 + l, \frac{1 + l}{2}} (Σ_{T})$ and we obtain the estimates (4.6), (4.7) for the functions $Φ_{1}$ and $Φ_{2}$ due to the estimates (4.19), (4.20) for the functions V and $\partial_{t} V$ . □

5. Unique solvability of the problem (1.2)–(1.4) and estimates of it’s solution

5.1. Solution of the problem (4.1)–(4.3) that is equivalent to the problem (1.2)–(1.4)

We have reduced the problem (1.2)–(1.4) to the equivalent one (4.1)–(4.3) for the homogeneous parabolic equation with unknown function $v (x, t)$ , but we shall study more general problem for non-homogeneous equation $\begin{array}{l} (5.1) & \partial_{t} z - A (x, t, \partial_{x}) z = g (x, t) in Ω_{T}, \\ (5.2) & z |_{t = 0} = 0 in Ω, \\ (5.3) & (ε \partial_{t} z - \overline{b} (x, t) \nabla^{T} z + b_{0} (x, t) z) |_{Σ} = Φ_{1} (x, t) + ε Φ_{2} (x, t), t \in (0, T) . \end{array}$

We shall prove the following theorem for this problem.

Theorem 5.1.
Let $Σ \in C^{2 + l}$ , l be positive non-integer, $0 < ε ⩽ ε_{0}$ , the conditions ( 1.8 ), ( 3.1 ) be fulfilled.

For any functions $g (x, t) \in {\overset{\circ}{C}}_{x t}^{l, l / 2} ({\overline{Ω}}_{T})$ , $Φ_{1} (x, t), Φ_{2} (x, t) \in {\overset{\circ}{C}}_{x t}^{1 + l, \frac{1 + l}{2}} (Σ_{T})$ , $j = 1, 2$ , the problem ( 5.1 )–( 5.3 ) has a unique solution $z (x, t) \in {\overset{\circ}{C}}_{x t}^{2 + l, 1 + l / 2} ({\overline{Ω}}_{T})$ , $ε \partial_{t} v (x, t) \in {\overset{\circ}{C}}_{x t}^{1 + l, \frac{1 + l}{2}} (Σ_{T})$ , and it satisfies an estimate $\begin{matrix} (5.4) & | z |_{Ω_{T}}^{(2 + l)} + | ε \partial_{t} z |_{Σ_{T}}^{(1 + l)} ⩽ C_{1} (| g |_{Ω_{T}}^{(l)} + | Φ_{1} |_{Σ_{T}}^{(1 + l)} + ε | Φ_{2} |_{Σ_{T}}^{(1 + l)}), \end{matrix}$ where the constant $C_{1}$ does not depend on ε.

In this theorem the given functions satisfy zero initial conditions, and the compatibility conditions are fulfilled automatically. Proof.
We derive an estimate (5.4) of the solution of the problem (5.1)–(5.3) with the help of the Schauder method. We cover the domain Ω with the balls $B_{κ, λ} (ξ_{κ})$ and $B_{κ, 2 λ} (ξ_{κ})$ with the radiuses $λ / 2$ and λ respectively and with the common center at the point $ξ_{κ}$ , where λ is a sufficiently small positive value.

These balls are enumerated as follows. For $κ \in N_{1}$ the balls $B_{κ, 2 λ} (ξ_{κ})$ lie entirely inside of the domain Ω; for $κ \in N_{2}$ the balls $B_{κ, λ} (ξ_{κ})$ intersect the boundary Σ: $B_{κ, λ} (ξ_{κ}) \cap Σ \neq \emptyset$ .

We construct the system of the smooth functions ${ζ_{κ} (x)}$ subordinated to this overlapping by the balls, such that $ζ_{κ} = 1$ , if $x \in B_{κ, λ} (ξ_{κ})$ and $ζ_{κ} = 0$ , if $x \in Ω ∖ B_{κ, 2 λ} (ξ_{κ})$ , and $| \partial^{m} ζ_{κ} | ⩽ C_{κ, m} λ^{- | m |}$ , $m = (m_{1}, \dots, m_{n})$ , $m_{i}$ are non-negative integers.

We multiply the right and the left – hand sides of the equation (5.1) and the conditions (5.2), (5.3) by $ζ_{κ} (x)$ . Depending on the position of the balls $B_{κ, 2 λ} (ξ_{κ})$ we shall have the Cauchy ( $κ \in N_{1}$ ) or boundary–value problem with a time derivative in the boundary condition ( $κ \in N_{2}$ ).

Let $κ \in N_{1}$ . We extend the functions $z_{κ} (x) : = ζ_{κ} (x) z (x)$ , $g_{κ} (x) : = ζ_{κ} (x) g (x, t)$ by zero outside of the ball $B_{κ, 2 λ} (ξ_{κ})$ into $R^{n}$ remaining their notations, then we obtain the Cauchy problem $\begin{matrix} (5.5) & \partial_{t} z_{κ} - A (x, t, \partial_{x}) z_{κ} = g_{κ} (x, t) + F_{κ}^{(1)} (x, t) in R_{T}^{n}, z_{κ} |_{t = 0} = 0 in R^{n}, \end{matrix}$ where $R_{T}^{n} : = R^{n} \times (0, T)$ , $\begin{matrix} (5.6) & F_{κ}^{(1)} = - \sum_{i, j = 1}^{n} a_{i j} (x, t) (z \partial_{x_{i} x_{j}}^{2} ζ_{κ} + 2 \partial_{x_{i}} z \partial_{x_{j}} ζ_{κ}) - z \sum_{i = 1}^{n} a_{i} (x, t) \partial_{x_{i}} ζ_{κ} . \end{matrix}$

The Cauchy problem (5.5) has unique solution [25] $z_{κ} (x, t) \in {\overset{\circ}{C}}_{x t}^{2 + l, 1 + l / 2} (R_{T}^{n})$ and it satisfies an estimate $\begin{matrix} (5.7) & | z_{κ} |_{R_{T}^{n}}^{(2 + l)} ⩽ C_{2} (| g_{κ} |_{R_{T}^{n}}^{(l)} + {| F_{κ}^{(1)} |}_{B_{R_{T}^{n}}}^{(l)}) . \end{matrix}$

Let $κ \in N_{2}$ . Without loss of generality we shall consider the case, when the centers $ξ_{κ}$ of the balls $B_{κ, λ} (ξ_{κ})$ are on the boundary Σ. We pass to the local coordinates system ${\overline{y}}, {\overline{y}} = {{\overline{y}}_{1}, \dots, {\overline{y}}_{n}}$ with the center in $ξ_{κ}$ and the axis ${\overline{y}}_{n}$ directed along the normal ${\overline{ν}}_{0} (ξ_{κ})$ to the surface Σ into Ω. We choose a radius λ sufficiently small so that the boundary $Σ \cap B_{κ, 2 λ}$ can be expressed by the equation ${\overline{y}}_{n} = μ_{κ} ({\overline{y}}^{'})$ , ${\overline{y}}^{'} \in d_{κ, 2 λ}$ , where $d_{κ, 2 λ}$ is a projection of a ball $B_{κ, 2 λ}$ on a hyperplane ${\overline{y}}_{n} = 0$ . As $Σ \in C^{2 + l}$ , then the function $μ_{κ} ({\overline{y}}^{'})$ belongs to $C^{2 + l} ({\overline{d}}_{κ, 2 λ})$ and there exists a constant $M > 0$ for all $κ \in N_{2}$ such that $\begin{matrix} (5.8) & | μ_{κ} |_{d_{κ, 2 λ}}^{(2 + l)} ⩽ M, μ_{κ} ({\overline{y}}^{'}) |_{{\overline{y}}^{'} = 0} = 0, \nabla^{'} μ_{κ} ({\overline{y}}^{'}) |_{{\overline{y}}^{'} = 0} = 0, | \partial_{{\overline{y}}_{i}} μ_{κ} ({\overline{y}}^{'}) | ⩽ M | {\overline{y}}^{'} |, \end{matrix}$ ${\overline{y}}^{'} \in d_{κ, 2 λ}$ , where $\nabla^{'} = (\partial_{{\overline{y}}_{1}}, \dots, \partial_{{\overline{y}}_{n - 1}})$ , $i = 1, \dots, n - 1$ .

We extend the function $μ_{κ}$ into $R^{n - 1}$ with preserving of it’s smoothness and keep it’s notation. Then we change the coordinates by the formulas $\begin{matrix} (5.9) & y^{'} = {\overline{y}}^{'}, y_{n} = {\overline{y}}_{n} - μ_{κ} ({\overline{y}}^{'}) . \end{matrix}$

Thus, we have done the coordinate transform $x = Y_{κ} (y)$ consisting of rotation around a point $x = ξ_{κ}$ and a mapping (5.9).

Let $D_{κ}$ be the half – space $y_{n} > 0$ , $D_{κ, T} = D_{κ} \times (0, T)$ , $R_{κ}$ be hyperplane $y_{n} = 0$ , $R_{κ, T} = R_{κ} \times [0, T]$ .

We denote $z_{κ}^{} (y, t) = ζ_{κ} (x) z (x, t) |_{x = Y_{κ} (y)}$ , $g_{κ}^{} (x, t) : = ζ_{κ} (x) g (x, t) |_{x = Y_{κ} (y)}$ , $Φ_{j, κ}^{} (y^{'}, t) = ζ_{κ} (x) \times Φ_{j, κ} (x, t) |_{x = Y_{κ} (y), y_{n} = 0}$ , and extend the functions $z_{κ}^{} (y, t)$ , $g_{κ}^{} (y, t)$ outside of the ball $B_{κ, 2 λ} (ξ_{κ})$ into $D_{κ}$ and $Φ_{ε, κ}^{} (y^{'}, t)$ outside of $d_{κ, 2 λ}$ into $R_{κ}$ by zero remaining their notations.

For the function $z_{κ}^{} (y, t)$ satisfying zero initial data we obtain the boundary–value problem $\begin{array}{l} (5.10) & \partial_{t} z_{κ}^{} - \sum_{i, j = 1}^{n} a_{i j}^{} (ξ_{κ}, 0) \partial_{y_{i} y_{j}}^{2} z_{κ}^{} = g_{κ}^{} (y, t) + F_{κ}^{(2)} in D_{κ, T}, \\ (5.11) & (ε \partial_{t} z_{κ}^{} - \sum_{i = 1}^{n} b_{i}^{} (ξ_{κ}, 0) \partial_{y_{i}} z_{κ}^{}) |_{y_{n} = 0} = Φ_{1, κ}^{} (y^{'}, t) + ε Φ_{2, κ}^{} (y^{'}, t) + P_{κ}, t \in (0, T), \end{array}$ where $a_{i j}^{} (ξ_{κ}, 0)$ , $b_{i}^{} (ξ_{κ}, 0)$ are the coefficients $a_{i j} (ξ_{κ}, 0)$ , $b_{i} (ξ_{κ}, 0)$ after the coordinate mapping $x = Y_{κ} (y)$ , $\begin{array}{l} F_{κ}^{(2)} = & ζ_{κ} (x) (\sum_{i, j = 1}^{n} (a_{i j} (x, t) - a_{i j} (ξ_{κ}, 0)) \partial_{x_{i} x_{j}}^{2} z + \sum_{i = 1}^{n} a_{i} (x, t) \partial_{x_{i}} z + a (x, y) z) |_{x = Y_{κ} (y)} \\ - \sum_{i, j = 1}^{n} a_{i j} (x, t) (z \partial_{x_{i} x_{j}}^{2} ζ_{κ} + 2 \partial_{x_{i}} z \partial_{x_{j}} ζ_{κ}) |_{x = Y_{κ} (y)} \\ + \sum_{i, j = 1}^{n - 1} a_{i j}^{} (ξ_{κ}, 0) \partial_{y_{i}} μ_{κ} \partial_{y_{j}} μ_{κ} \partial_{y_{n}}^{2} z_{κ}^{} - 2 \sum_{i = 1}^{n - 1} \sum_{j = 1}^{n} a_{i j}^{} (ξ_{κ}, 0) \partial_{y_{i}} μ_{κ} \partial_{y_{j} y_{n}}^{2} z_{κ}^{} \\ (5.12) & - \sum_{i, j = 1}^{n - 1} a_{i j}^{} (ξ_{κ}, 0) \partial_{y_{i} y_{j}}^{2} μ_{κ} \partial_{y_{n}} z_{κ}^{}, \\ P_{κ} = & (ζ_{κ} (x) ((\overline{b} (x, t) - \overline{b} (ξ_{κ}, 0)) \nabla^{T} z - b_{0} (x, t) z) - z \overline{b} (ξ_{κ}, 0) \nabla^{T} ζ_{κ}) |_{\begin{array}{c} x = Y_{κ} (y), \\ y_{n} = 0 \end{array}} \\ (5.13) & - \sum_{i = 1}^{n - 1} b_{i}^{} (ξ_{κ}, 0) \partial_{y_{i}} μ_{κ} \partial_{y_{n}} z_{κ}^{} |_{y_{n} = 0} . \end{array}$ We point out that after the rotation of a coordinate system (i.e. orthogonal transform) the matrix ${a_{i j}^{} (ξ_{κ}, 0)}_{i, j = 1}^{n}$ has the same positive eigenvalues as original matrix ${a_{i j} (x, t)}_{i, j = 1}^{n}$ at the point $(ξ_{κ}, 0)$ , that is an equation (5.10) is parabolic.

The coefficient $b_{n}^{} (ξ_{κ}, 0)$ in the boundary condition (5.11) is positive. Really, we have the condition (1.8): $\overline{b} (x, t) {\overline{ν}}^{T} (x) ⩾ d_{0} > 0$ $\forall (x, t) \in Σ_{T}$ . After rotation of a coordinate system scalar product $\overline{b} (ξ_{κ}, 0) {\overline{ν}}^{T} (ξ_{κ})$ is not changed and $\overline{b} (ξ_{κ}, 0) {\overline{ν}}^{T} (ξ_{κ}) = {\overline{b}}^{} (ξ_{κ}, 0) {\overline{ν}}^{ T} (ξ_{κ}) = b_{n}^{} (ξ_{κ}, 0) > 0$ , here $\overline{ν} (ξ_{k})$ and ${\overline{ν}}^{} (ξ_{k})$ are the unit normals to the boundary Σ and plane $y_{n} = 0$ directed into Ω and $D_{κ}$ respectively.

(5.10), (5.11) is a model problem (A.1)–(A.3) for which the Theorem A.1 is proved. In according to it the problem (5.10), (5.11) has a unique solution $z_{κ}^{} (y, t) \in {\overset{\circ}{C}}_{y t}^{2 + l, 1 + l / 2} ({\overline{D}}_{κ, T})$ , $ε \partial_{t} v_{κ}^{} (y, t) \in {\overset{\circ}{C}}_{y t}^{1 + l, \frac{1 + l}{2}} (R_{κ, T})$ , and it satisfies an estimate $\begin{matrix} (5.14) & \begin{matrix} {| z_{κ}^{} |}_{D_{κ, T}}^{(2 + l)} + {| ε \partial_{t} z_{κ}^{} |}_{R_{κ, T}}^{(1 + l)} \\ ⩽ C_{3} ({| g_{κ}^{} |}_{D_{κ, T}}^{(l)} + {| F_{κ}^{(2)} |}_{D_{κ, T}}^{(l)} + {| Φ_{1, κ}^{} |}_{R_{κ, T}}^{(1 + l)} + ε {| Φ_{2, κ}^{} |}_{R_{κ, T}}^{(1 + l)} + | P_{κ} |_{R_{κ, T}}^{(1 + l)}), \end{matrix} \end{matrix}$ where the constant $C_{3}$ does not depend on ε.

Now in (5.7) and (5.14) we evaluate the norms of the functions $F_{κ}^{(1)}$ , $F_{κ}^{(2)}$ , $P_{κ}$ determined by the formulas (5.6), (5.12), (5.13) with the help of (5.8) and the inequalities $\begin{matrix} {| \partial_{t}^{k} \partial_{x}^{m} u |}_{Ω_{T}} ⩽ ϵ^{l - 2 k - | m |} {[u]}_{Ω_{T}}^{(l)} + C ϵ^{- 2 k - | m |} | u |_{Ω_{T}}, \\ {[u]}_{Ω_{T}}^{(s)} ⩽ ϵ^{l - s} {[u]}_{Ω_{T}}^{(l)} + C ϵ^{- s} | u |_{Ω_{T}}, ϵ > 0, \end{matrix}$ here $ϵ \in (0, 1)$ , $0 < 2 k + | m | < l$ , $0 < s < l$ , C is a positive constant. Choosing λ, $t \in (0, t_{1}]$ and ϵ sufficiently small we obtain $\begin{array}{l} (5.15) & | z_{κ} |_{R_{t_{1}}^{n}}^{(2 + l)} ⩽ C_{4} | g_{κ} |_{R_{t_{1}}^{n}}^{(l)} + δ | z_{κ} |_{R_{t_{1}}^{n}}^{(2 + l)}, κ \in N_{1}, \\ (5.16) & \begin{matrix} {| z_{κ}^{} |}_{D_{κ, T}}^{(2 + l)} + {| ε \partial_{t} z_{κ}^{} |}_{R_{κ, T}}^{(1 + l)} ⩽ & C_{5} ({| g_{κ}^{} |}_{D_{κ, T}}^{(l)} + {| Φ_{1, κ}^{} |}_{R_{κ, t_{1}}}^{(1 + l)} + ε | Φ_{2, κ}^{} |_{R_{κ, t_{1}}}^{(1 + l)} |) \\ + δ {| z_{κ}^{} |}_{D_{κ, T}}^{(2 + l)}, κ \in N_{2}, \end{matrix} \end{array}$ where $δ = δ (t_{1}, λ, ϵ) \in (0, 1)$ .

We take into account that $z_{κ}^{} (y, t) |_{y = Y_{κ}^{- 1}} (x) = z_{κ} (x, t)$ and thanks to the cut-off function $ζ_{κ} (x)$ we have $z_{κ} (x, t) : = ζ_{κ} (x) z (x, t) = z (x, t)$ , $g_{κ} (x, t) : = ζ_{κ} (x) g (x, t) = g (x, t)$ in $B_{κ, λ}$ , $Φ_{j, κ} (x, t) = Φ_{j} (x, t)$ , $j = 1, 2$ , in $d_{κ, λ}$ . Let $B_{κ, λ, t} : = B_{κ, λ} \times (0, t)$ , $d_{κ, λ, t_{1}} = d_{κ, λ} \times (0, t_{1})$ , then we can write the estimates (5.15), (5.16) in the form $\begin{array}{l} (5.17) & | z |_{B_{κ, λ, t}}^{(2 + l)} ⩽ C_{6} | g |_{B_{κ, λ, t}}^{(l)} + δ | z |_{B_{κ, λ, t}}^{(2 + l)}, κ \in N_{1}, \\ (5.18) & \begin{matrix} | z |_{B_{κ, t}}^{(2 + l)} + | ε \partial_{t} z |_{d_{κ, λ, t}}^{(1 + l)} \\ ⩽ C_{7} (| g |_{B_{κ, λ, t}}^{(l)} + | Φ_{1} |_{d_{κ, λ, t}}^{(1 + l)} + ε | Φ_{2} |_{d_{κ, λ, t}}^{(1 + l)}) + δ | z |_{B_{κ, λ, t}}^{(2 + l)}, κ \in N_{2} . \end{matrix} \end{array}$

We find the inequalities from (5.17), (5.18) for $t \in [0, t_{1}]$ $\begin{array}{l} (5.19) & \begin{array}{l} | z |_{B_{κ, λ, t}}^{(2 + l)} ⩽ C_{8} | g |_{B_{κ, λ, t}}^{(l)}, κ \in N_{1}, \\ | z |_{B_{κ, t}}^{(2 + l)} + | ε \partial_{t} z |_{d_{κ, λ, t}}^{(1 + l)} ⩽ C_{9} (| g |_{B_{κ, λ, t}}^{(l)} + | Φ_{1} |_{d_{κ, λ, t}}^{(1 + l)} + ε | Φ_{2} |_{d_{κ, λ, t}}^{(1 + l)}), κ \in N_{2}, \end{array} \end{array}$ and in the inequalities (5.19) we pass to the maximum with respect to κ and t $\begin{array}{l} max_{\begin{array}{c} κ \in N_{1} \cup N_{2}, \\ t \in [0, t_{1}] \end{array}} | z |_{B_{κ, λ, t}}^{(2 + l)} + max_{\begin{array}{c} κ \in N_{2}, \\ t \in [0, t_{1}] \end{array}} | ε \partial_{t} z |_{d_{κ, λ, t}}^{(1 + l)} \\ (5.20) & ⩽ C_{10} (max_{\begin{array}{c} κ \in N_{1} \cup N_{2}, \\ t \in [0, t_{1}] \end{array}} | g |_{B_{κ, λ, t}}^{(l)} + max_{\begin{array}{c} κ \in N_{2}, \\ t \in [0, t_{1}] \end{array}} | Φ_{1} |_{d_{κ, λ, t}}^{(1 + l)} + ε max_{\begin{array}{c} κ \in N_{2}, \\ t \in [0, t_{1}] \end{array}} | Φ_{2} |_{d_{κ, λ, t}}^{(1 + l)}) . \end{array}$

The norm ${max}_{\begin{array}{c} κ \in N_{1} \cup N_{2}, \\ t \in [0, t_{1}] \end{array}} | f |_{B_{κ, λ, t}}^{(l)}$ is equivalent to the norm $| f |_{Ω_{t_{1}}}^{(l)}$ [25], Ch.IV, so we can write an estimate (5.20) as follows: $\begin{matrix} (5.21) & | z |_{Ω_{t}}^{(2 + l)} + | ε \partial_{t} z |_{Σ_{t}}^{(1 + l)} ⩽ C_{11} (| g |_{Ω_{t}}^{(l)} + | Φ_{1} |_{Σ_{t}}^{(1 + l)} + ε | Φ_{2} |_{Σ_{t}}^{(1 + l)}), t \in (0, t_{1}], \end{matrix}$ where the constant $C_{11}$ does not depend on ε.

The existence of the solution of the problem (5.1)–(5.3) for small $t \in (0, t_{2}]$ is proved with the help of construction of regularizer $R h = Σ_{κ} η_{κ} (x) z_{κ} (x, t)$ [25], Ch.IV, where $\overline{h} = {g, Φ_{1}, ε Φ_{2}}$ is given vector – function, $z_{κ}$ is the solutions of the Cauchy problem, $κ \in N_{1}$ , and a model boundary – value problem as (A.1)–(A.3) for $κ \in N_{2}$ , $η_{κ} (x)$ is a smooth function, such that $Σ_{κ} ζ_{κ} (x) η_{κ} (x) = 1$ .

We substitute function $R \overline{h}$ into the equation and conditions of the problem (5.1)–(5.3) and obtain the problems as (5.5) for $κ \in N_{1}$ , and (5.10), (5.11) for $κ \in N_{2}$ with the operators similar to the operators in (5.6) and (5.12), (5.13). We estimate them and then with the help of the contraction mapping principle we prove an existence of the solution of the problem (5.1)–(5.3) for small $t \in (0, t_{2}]$ .

Let $t_{0} = min (t_{1}, t_{2})$ . We extend the solution from $[0, t_{0}]$ into $[0, T]$ as in [25], Ch.IV, §8, and obtain Theorem 4.1 and an estimate (5.21) for $t \in (0, T]$ .

The estimate of the solution leads to it’ uniqueness. □

5.2. Proof of Theorem 3.1

We have reduced the problem (1.2)–(1.4) with unknown function $u (x, t)$ to the equivalent one (4.1)–(4.3) for the homogeneous equation and with unknown function $v (x, t)$ . (4.1)–(4.3) is the problem (5.1)–(5.3) with $g = 0$ . From the Theorem 5.1 for the problem (5.1)–(5.3) it follows that the problem (4.1)–(4.3) has unique solution $v (x, t) \in {\overset{\circ}{C}}_{x t}^{2 + l, 1 + l / 2} ({\overline{Ω}}_{T})$ , $ε \partial_{t} v (x, t) \in {\overset{\circ}{C}}_{x t}^{1 + l, \frac{1 + l}{2}} (Σ_{T})$ , and $\begin{matrix} | v |_{Ω_{T}}^{(2 + l)} + | ε \partial_{t} v |_{Σ_{T}}^{(1 + l)} ⩽ C_{12} (| Φ_{1} |_{Σ_{T}}^{(1 + l)} + ε | Φ_{2} |_{Σ_{T}}^{(1 + l)}), \end{matrix}$ where the constant $C_{12}$ does not depend on ε.

We remember the substitution (4.21): $u (x, t) = v (x, t) + V (x, t)$ , where V is the solution of the problem (4.14)–(4.16), such that $V (x, t) \in C_{x t}^{2 + l, 1 + l / 2} (Ω_{T})$ , $\partial_{t} V (x, t) |_{Σ} \in C_{x t}^{1 + l, \frac{1 + l}{2}} (Σ_{T})$ and these functions satisfy the estimates (4.17), (4.20): $\begin{matrix} | V |_{Ω_{T}}^{(2 + l)} ⩽ C_{13} (| u_{0} |_{Ω}^{(2 + l)} + | f |_{Ω_{T}}^{(l)}), | \partial_{t} V |_{Σ_{T}}^{(1 + l)} ⩽ C_{14} \sum_{p = 0}^{[\frac{1 + l}{2}]} | F_{p} |_{Σ}^{(1 + l - 2 p)} . \end{matrix}$

Function $u (x, t)$ is the solution of the original problem (1.2)–(1.4) and from above we shall have that $u = v + V \in C_{x t}^{2 + l, 1 + l / 2} (Ω_{T})$ , $ε \partial_{t} u |_{Σ} \in C_{x t}^{1 + l, \frac{1 + l}{2}} (Σ_{T})$ and $\begin{array}{l} | u |_{Ω_{T}}^{(2 + l)} + | ε \partial_{t} u |_{Σ_{T}}^{(1 + l)} \\ ⩽ C_{15} (| u_{0} |_{Ω}^{(2 + l)} + | f |_{Ω_{T}}^{(l)} + | Φ_{1} |_{Σ_{T}}^{(1 + l)} + C_{16} ε (| Φ_{2} |_{Σ_{T}}^{(1 + l)} + \sum_{p = 0}^{[\frac{1 + l}{2}]} | F_{1 + p} |_{Σ}^{(1 + l - 2 p)})), \end{array}$ where the constants $C_{15}$ , $C_{16}$ do not depend on ε.

We apply here the estimates (4.6), (4.7) of the functions $Φ_{1}$ , $ε Φ_{2}$ , then we obtain an estimate (3.3) and Theorem 3.1.

5.3. Proof of Theorem 3.3

We shall derive an estimate (3.4) of the time derivative $ε \partial_{t} u |_{Σ_{T}}$ .

Consider the problem (5.1)–(5.3) with unknown function $z (x, t)$ . In the proof of the estimate (5.4) for $z (x, t)$ with the Schauder method we have obtained the problem (5.10), (5.11) with unknown function $z_{κ}^{*} (y, t), κ \in N_{2}$ . (5.10), (5.11) is a model problem (A.1)–(A.3), for which Theorem A.2 with an estimate (A.5) is proved. All the conditions of Theorem A.2 are fulfilled, so we make use of an estimate (A.5) for the time derivative $ε \partial_{t} z_{κ}^{*}$ $\begin{matrix} {| ε \partial_{t} z_{κ}^{*} |}_{R_{κ, T}}^{(1 + k + β)} ⩽ C_{17} ε^{\frac{α - β}{2}} ({| g_{κ}^{*} + F_{κ}^{(2)} |}_{D_{κ, T}}^{(k + α)} + {| Φ_{1, κ}^{*} + ε Φ_{2, κ}^{*} + P_{κ} |}_{R_{κ, T}}^{(1 + k + α)}), \end{matrix}$ where $β \in (0, α / 2)$ , constant $C_{17}$ does not depend on ε, the functions $F_{κ}^{(2)}$ , $P_{κ}$ are determined by the formulas (5.12), (5.13), they depend on $z_{κ}^{*}$ and are known by the Theorem 5.1.

By direct evaluation of the norms $| F_{κ}^{(2)} |_{D_{κ, T}}^{(k + α)}$ , $| P_{κ} |_{R_{κ, T}}^{(1 + k + α)}$ with the help of the estimate (5.16) for $z_{κ}^{*}$ we find $\begin{matrix} {| ε \partial_{t} z_{κ}^{*} |}_{R_{κ, T}}^{(1 + k + β)} ⩽ C_{18} ε^{\frac{α - β}{2}} ({| g_{κ}^{*} |}_{D_{κ, T}}^{(k + α)} + {| Φ_{1, κ}^{*} |}_{R_{κ, T}}^{(1 + k + α)} + {| ε Φ_{2, κ}^{*} |}_{R_{κ, T}}^{(1 + k + α)} + δ {| z_{κ}^{*} |}_{R_{κ, T}}^{(2 + k + α)}) . \end{matrix}$

We rewrite this estimate as (5.18) $\begin{matrix} | ε \partial_{t} z |_{d_{κ, λ, t}}^{(1 + k + β)} ⩽ C_{19} ε^{\frac{α - β}{2}} (| g |_{B_{κ, λ, t}}^{(k + α)} + | Φ_{1} |_{d_{κ, λ, t}}^{(1 + k + α)} + | ε Φ_{2} |_{d_{κ, λ, t}}^{(1 + k + α)} + δ | z |_{κ, λ, t}^{(2 + k + α)}) \end{matrix}$ and apply an estimate (5.19) with $l = k + α$ for $| z |_{B_{κ, λ, t}}^{(2 + k + α)}$ , then we shall have $\begin{matrix} | ε \partial_{t} z |_{d_{κ, λ, t}}^{(1 + k + β)} ⩽ C_{20} ε^{\frac{α - β}{2}} (| g |_{B_{κ, λ, t}}^{(k + α)} + | Φ_{1} |_{d_{κ, λ, t}}^{(1 + k + α)} + ε | Φ_{2} |_{d_{κ, λ, t}}^{(1 + k + α)}), κ \in N_{2} . \end{matrix}$

We take maximum with respect to κ and $t \in [0, t_{3}]$ in this inequality $\begin{array}{l} max_{\begin{array}{c} κ \in N_{2}, \\ t \in [0, t_{3}] \end{array}} | ε \partial_{t} z |_{d_{κ, λ, t}}^{(1 + k + β)} \\ (5.22) & ⩽ C_{21} ε^{\frac{α - β}{2}} (max_{\begin{array}{c} κ \in N_{2}, \\ t \in [0, t_{3}] \end{array}} | g |_{B_{κ, λ, t}}^{(k + α)} + max_{\begin{array}{c} κ \in N_{2}, \\ t \in [0, t_{3}] \end{array}} | Φ_{1} |_{d_{κ, λ, t}}^{(1 + k + α)} + ε max_{\begin{array}{c} κ \in N_{2}, \\ t \in [0, t_{3}] \end{array}} | Φ_{2} |_{d_{κ, λ, t}}^{(1 + k + α)}) \end{array}$ and rewrite (5.22) due to an equivalence of the norms ${max}_{\begin{array}{c} κ \in N_{2}, \\ t \in [0, t_{3}] \end{array}} | f |_{B_{κ, λ, t}}^{(l)}$ and $| f |_{Ω_{t_{3}}}^{(l)}$ [25] $\begin{matrix} (5.23) & | ε \partial_{t} z |_{Σ_{t}}^{(1 + k + α)} ⩽ C_{22} ε^{\frac{α - β}{2}} (| g |_{Ω_{t}}^{(k + α)} + | Φ_{1} |_{Σ_{t}}^{(1 + k + α)} + ε | Φ_{2} |_{Σ_{t}}^{(1 + k + α)}), t \in (0, t_{3}] . \end{matrix}$

We extend an estimate (5.23) into $[0, T]$ as above and obtain an estimate (5.23) for $t \in [0, T]$ .

We return to the problem (4.1)–(4.3) with an unknown function $v (x, t)$ . (4.1)–(4.3) is the problem (5.1)–(5.3) with $g = 0$ , so for the derivative $ε \partial_{t} v$ we shall have an estimate (5.23) with $g = 0$ $\begin{matrix} (5.24) & | ε \partial_{t} v |_{Σ_{t}}^{(1 + k + β)} ⩽ C_{23} ε^{\frac{α - β}{2}} (| Φ_{1} |_{Σ_{t}}^{(1 + k + α)} + ε | Φ_{2} |_{Σ_{t}}^{(1 + k + α)}), t \in (0, T] . \end{matrix}$

Now we consider the original problem (1.2)–(1.4) with unknown function $u (x, t)$ . We have done the substitution (4.21): $u (x, t) = v (x, t) + V (x, t)$ , where V is the solution of the problem (4.14)–(4.16). We make use of the estimates (5.24) for $ε \partial_{t} v$ , (4.8) for $\partial_{t} V$ , then we shall have $\begin{array}{l} | ε \partial_{t} u |_{Σ_{T}}^{(1 + k + β)} & ⩽ | ε \partial_{t} v |_{Σ_{T}}^{(1 + k + β)} | + ε | \partial_{t} V |_{Σ_{T}}^{(1 + k + β)} \\ ⩽ | ε \partial_{t} v |_{Σ_{T}}^{(1 + k + β)} | + C_{24} ε | \partial_{t} V |_{Σ_{T}}^{(1 + k + α)} \\ ⩽ C_{25} ε^{\frac{α - β}{2}} (| Φ_{1} |_{Σ_{t}}^{(1 + k + α)} + ε | Φ_{2} |_{Σ_{t}}^{(1 + k + α)}) + C_{26} ε \sum_{p = 0}^{[\frac{1 + k}{2}]} | F_{p} |_{Σ}^{(1 + k + α - 2 p)} . \end{array}$

Using the estimates (4.6), (4.7) for $Φ_{1}$ , $Φ_{2}$ we obtain the inequality for $\forall ε \in (0, ε_{0}]$ $\begin{array}{l} | ε \partial_{t} u |_{Σ_{T}}^{(1 + k + β)} ⩽ & C_{27} ε^{\frac{α - β}{2}} (| u_{0} |_{Ω}^{(2 + k + α)} + | f |_{Ω_{T}}^{(k + α)} + | φ_{1} |_{Σ_{T}}^{(1 + k + α)}) \\ + C_{28} ε (ε_{0}^{\frac{α - β}{2}} | φ_{2} |_{Σ_{T}}^{(1 + k + α)} + (1 + ε_{0}^{\frac{α - β}{2}}) \sum_{p = 0}^{[\frac{1 + k}{2}]} | F_{1 + p} |_{Σ}^{(1 + k + α - 2 p)}) \\ ⩽ & C_{27} ε^{\frac{α - β}{2}} (| u_{0} |_{Ω}^{(2 + k + α)} + | f |_{Ω_{T}}^{(k + α)} + | φ_{1} |_{Σ_{T}}^{(1 + k + α)}) \\ (5.25) & + C_{29} ε (| φ_{2} |_{Σ_{T}}^{(1 + k + α)} + \sum_{p = 0}^{[\frac{1 + k}{2}]} | F_{1 + p} |_{Σ}^{(1 + k + α - 2 p)}), \end{array}$ where the constants $C_{27}$ , $C_{29}$ are independent on ε. (5.25) is the required estimate (3.4).

6. Convergence of the solution of the problem (1.2)–(1.4) to the solution of the problem (1.5)–(1.7)

Here we shall make use of the letter p and indexes n, q in $ε_{n}$ , $ε_{n_{q}}$ , although these letters were applied in other meanings.

The convergence of the solution of the problem (1.2)–(1.4) is proved with the help of the compact imbedding of the Hölder spaces. This method was applied in [2, 5].

6.1. Proof of Theorem 3.4

Consider the original problem (1.2)–(1.4) with unknown function $u (x, t)$ . We designate $u (x, t) = : u_{ε} (x, t)$ . We should show that $u_{ε} (x, t)$ converges as $ε \to 0$ to the solution $w (x, t)$ of the unperturbed problem (1.5)–(1.7).

Let $k = 0, 1, \dots, α \in (0, 1)$ .

When $ε \to 0$ , we have the sequence ${u_{ε} (x, t)}$ of the solutions of the problem (1.2)–(1.4). By Theorem 3.1 the problem (1.2)–(1.4) has unique solution, such that $u_{ε} (x, t) \in C_{x t}^{2 + k + α, 1 + \frac{k + α}{2}} ({\overline{Ω}}_{T})$ , $ε \partial_{t} u_{ε} |_{Σ} \in C_{x t}^{1 + k + α, \frac{1 + k + α}{2}} (Σ_{T})$ and an estimate for them holds for all $ε \in (0, ε_{0}]$ $\begin{matrix} (6.1) & | u_{ε} |_{Ω_{T}}^{(2 + k + α)} + | ε \partial_{t} u_{ε} |_{Σ_{T}}^{(1 + k + α)} ⩽ C_{1} M_{1} + ε C_{2} M_{2} ⩽ C_{1} M_{1} + ε_{0} C_{2} M_{2}, \end{matrix}$ where $\begin{array}{l} M_{1} = | u_{0} |_{Ω}^{(2 + k + α)} + | f |_{Ω_{T}}^{(k + α)} + | φ_{1} |_{Σ_{T}}^{(1 + k + α)}, \\ M_{2} = | φ_{2} |_{Σ_{T}}^{(1 + k + α)} + \sum_{p = 0}^{[\frac{1 + k}{2}]} | F_{1 + p} |_{Σ}^{(1 + k + α - 2 p)}, \end{array}$ the constants $C_{1}$ , $C_{2}$ do not depend on ε.

The set of the functions ${u_{ε} (x, t)}$ is bounded in $C_{x t}^{2 + k + α, 1 + \frac{k + α}{2}} ({\overline{Ω}}_{T})$ and compact in $C_{x t}^{2 + k + β, 1 + \frac{k + β}{2}} ({\overline{Ω}}_{T})$ , $β \in (0, α / 2)$ .

Let ${u_{ε_{n}}} \subset {u_{ε}}$ be converging subsequence in $C_{x t}^{2 + k + β, 1 + \frac{k + β}{2}} ({\overline{Ω}}_{T})$ as $n \to \infty (ε_{n} \to 0)$ . We denote $\begin{matrix} (6.2) & lim_{n \to \infty} u_{ε_{n}} (x, t) = : w (x, t) \in C_{x t}^{2 + k + β, 1 + \frac{k + β}{2}} ({\overline{Ω}}_{T}) . \end{matrix}$

From the estimate (6.1) it follows that $\begin{matrix} | u_{ε_{n}} |_{Ω_{T}}^{(2 + k + β)} ⩽ | u_{ε_{n}} |_{Ω_{T}}^{(2 + k + β)} + | ε \partial_{t} u_{ε_{n}} |_{Σ_{T}}^{(1 + k + β)} ⩽ C_{1} M_{1} + C_{2} ε_{n} M_{2} . \end{matrix}$

By the convergence of the subsequence $u_{ε_{n}} (x, t)$ to $w (x, t)$ as $n \to \infty$ we pass to the limit as $n \to \infty$ in this inequality and obtain $\begin{matrix} (6.3) & | w |_{Ω_{T}}^{(2 + k + β)} ⩽ C_{1} M_{1} . \end{matrix}$

The sequence ${ε_{n} \partial_{t} u_{ε_{n}} |_{Σ}}$ is bounded in $C_{x t}^{1 + k + α, \frac{1 + k + α}{2}} (Σ_{T})$ and compact in $C_{x t}^{1 + k + β, \frac{1 + k + β}{2}} (Σ_{T})$ , $β \in (0, α / 2)$ .

Let ${ε_{n_{q}} \partial_{t} u_{ε_{n_{q}}} |_{Σ}} \subset {ε_{n} \partial_{t} u_{ε_{n}} |_{Σ}}$ be converging subsequence in $C_{x t}^{1 + k + β, \frac{1 + k + β}{2}} (Σ_{T})$ .

In the Theorem 3.3 we have derived an estimate (3.4), which we write with $ε = ε_{n_{q}}$ $\begin{matrix} | ε_{n_{q}} \partial_{t} u_{n_{q}} |_{Σ_{T}}^{(1 + k + β)} ⩽ C_{3} ε_{n_{q}}^{\frac{α - β}{2}} M_{1} + C_{4} ε_{n_{q}} M_{2}, \end{matrix}$ where the constants $C_{3}$ , $C_{4}$ do not depend on $ε_{n_{q}}$ , and we obtain $\begin{matrix} (6.4) & lim_{q \to \infty} | ε_{n_{q}} \partial_{t} u_{ε_{n_{q}}} |_{Ω_{T}}^{(1 + k + β)} = 0 . \end{matrix}$

We have found the converging subsequence ${u_{ε_{n}}}$ as $n \to \infty$ . From it we choose the subsequence ${u_{ε_{n_{q}}}}$ with the same indexes as in the set ${ε_{n_{q}} \partial_{t} u_{ε_{n_{q}}} |_{Σ}}$ . The subsequence ${u_{ε_{n_{q}}}}$ is also converging to the function w as the subsequence of the converging sequence ${u_{ε_{n}}}$ , and we shall have by (6.2), (6.3) $\begin{matrix} (6.5) & lim_{q \to \infty} u_{ε_{n_{q}}} (x, t) = w (x, t), | w |_{Ω_{T}}^{(2 + k + β)} ⩽ C_{1} M_{1} . \end{matrix}$

We rewrite the problem (1.2)–(1.4) with $ε_{n_{q}}$ , $u_{ε_{n_{q}}}$ instead of ε and u, next, in this problem we pass to the limit as $q \to \infty$ taking into consideration (6.4), (6.5), then we obtain that the limit function $w (x, t)$ is the solution of the unperturbed problem (1.5)–(1.7), w belongs to the space $C_{x t}^{2 + k + β, 1 + \frac{k + β}{2}} ({\overline{Ω}}_{T}), β \in (0, α / 2)$ , and satisfies an estimate (6.5).

We shall prove that $w (x, t) \in C_{x t}^{2 + k + α, 1 + \frac{k + α}{2}} ({\overline{Ω}}_{T})$ .

We have due to the estimate (6.5) $\begin{matrix} (6.6) & | w |_{Ω_{T}}^{(2 + k)} ⩽ C_{1} M_{1} . \end{matrix}$

We shall estimate the Hölder constants. We evaluate, for example, the following Hölder constant. We consider the ratio with $(x, t), (x, t_{1}) \in {\overline{Ω}}_{T}$ , $2 m_{0} + | m | = 1 + k$ , $\begin{array}{l} (6.7) & \begin{array}{l} \begin{matrix} \frac{| \partial_{t}^{m_{0}} \partial_{x}^{m} w (x, t) - \partial_{t_{1}}^{m_{0}} \partial_{x}^{m} w (x, t_{1}) |}{| t - t_{1} |^{\frac{1 + α}{2}}} ⩽ & \frac{| \partial_{t}^{m_{0}} \partial_{x}^{m} w (x, t) - \partial_{t}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}} (x, t) |}{| t - t_{1} |^{\frac{1 + α}{2}}} \\ + \frac{| \partial_{t_{1}}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}} (x, t_{1}) - \partial_{t_{1}}^{m_{0}} \partial_{x}^{m} w (x, t_{1}) |}{| t - t_{1} |^{\frac{1 + α}{2}}} + I_{ε_{n}}, \end{matrix} \\ I_{ε_{n_{q}}} : = \frac{| \partial_{t}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}} (x, t) - \partial_{t_{1}}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}} (x, t_{1}) |}{| t - t_{1} |^{\frac{1 + α}{2}}} . \end{array} \end{array}$

We apply an estimate (6.1) for $u_{ε}$ , which is valid also for $u_{ε_{n_{q}}}$ , then $\begin{matrix} I_{ε_{n_{q}}} ⩽ sup_{(x, t), (x, t_{1}) \in {\overline{Ω}}_{T}} I_{ε_{n_{q}}} = {[\partial_{t}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}}]}_{t, Ω_{T}}^{(\frac{1 + α}{2})} ⩽ C_{1} M_{1} + ε_{n_{q}} C_{2} M_{2} \end{matrix}$ and $\begin{matrix} (6.8) & lim_{q \to \infty} I_{ε_{n_{q}}} ⩽ C_{1} M_{1} . \end{matrix}$

Further, by the convergence of the sequence ${u_{ε_{n_{q}}} (x, t)}$ to the function $w (x, t)$ in $C_{x t}^{2 + k + β, 1 + \frac{k + β}{2}} ({\overline{Ω}}_{T})$ we have $\begin{matrix} (6.9) & \partial_{t}^{m_{0}} \partial_{x}^{m} w (x, t) - \partial_{t}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}} (x, t) \to 0, \partial_{t_{1}}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}} (x, t) - \partial_{t_{1}}^{m_{0}} \partial_{x}^{m} w (x, t_{1}) \to 0 \end{matrix}$ as $q \to \infty$ in ${\overline{Ω}}_{T}$ .

We pass to the limit as $q \to \infty$ in the inequality (6.7) considering (6.8), (6.9) and take supremum with respect to $(x, t), (x, t_{1}) \in {\overline{Ω}}_{T}$ , then we shall have $\begin{matrix} (6.10) & sup_{(x, t), (x, t_{1}) \in {\overline{Ω}}_{T}} \frac{| \partial_{t}^{m_{0}} \partial_{x}^{m} w (x, t) - \partial_{t}^{m_{0}} \partial_{x}^{m} w (x, t_{1}) |}{| t - t_{1} |^{\frac{1 + α}{2}}} = {[\partial_{t}^{m_{0}} \partial_{x}^{m} w]}_{t, Ω_{T}}^{(\frac{1 + α}{2})} ⩽ C_{1} M_{1} . \end{matrix}$

In the same way we find the estimates of all other Hölder constants. Applying (6.6) and the estimates of the Hölder constants we obtain that $w (x, t) \in C_{x t}^{2 + k + α, 1 + \frac{k + α}{2}} ({\overline{Ω}}_{T})$ and $\begin{matrix} | w |_{Ω_{T}}^{(2 + k + α)} ⩽ C_{1} M_{1} = C_{1} (| u_{0} |_{Ω}^{(2 + k + α)} + | f |_{Ω_{T}}^{(k + α)} + | φ_{1} |_{Σ_{T}}^{(1 + k + α)}) . \end{matrix}$

The uniqueness of the solution $w (x, t)$ of the unperturbed problem (1.5)–(1.7) follows because of this estimate and due to the uniqueness of w we conclude that the whole sequence ${u_{ε} (x, t)}$ converges to the function $w (x, t)$ as $ε \to 0$ . □

6.2. Proof of Theorem 3.5

Consider the perturbed problem (1.2)–(1.4) with unknown function $u (x, t) : = u_{ε} (x, t)$ .

We set also for the obviousness $l = k + α, k = 0, 1, \dots, α \in (0, 1)$ .

From the Theorem 3.1 under the assumptions of the Theorem 3.5 it follows that the problem (1.2)–(1.4) has a unique solution $u_{ε} (x, t) \in C_{x t}^{3 + k + α, \frac{3 + k + α}{2}} ({\overline{Ω}}_{T})$ , $ε \partial_{t} u_{ε} (x, t) \in C_{x t}^{2 + k + α, 1 + \frac{k + α}{2}} (Σ_{T})$ , and it satisfies the estimate $\begin{array}{l} | u_{ε} |_{Ω_{T}}^{(3 + k + α)} + | ε \partial_{t} u_{ε} |_{Σ_{T}}^{(2 + k + α)} ⩽ C_{5} M_{3} + ε C_{6} M_{4}, \\ (6.11) & M_{3} : = (| u_{0} |_{Ω}^{(3 + k + α)} + | f |_{Ω_{T}}^{(1 + k + α)} + | φ_{1} |_{Σ_{T}}^{(2 + k + α)}), \\ M_{4} : = (| φ_{2} |_{Σ_{T}}^{(2 + k + α)} + \sum_{p = 0}^{1 + [k / 2]} | F_{1 + p} |_{Σ}^{(2 + k + α - 2 p)}), \end{array}$ where the constants $C_{5}$ , $C_{6}$ do not depend on ε.

We shall prove that the solution $u_{ε} (x, t)$ of the problem (1.2)–(1.4) converges as $ε \to 0$ to the unique solution $w (x, t) \in C_{x t}^{2 + k + γ, 1 + k / 2 + γ / 2} ({\overline{Ω}}_{T})$ of the problem (1.5)–(1.7), and an estimate takes place $\begin{matrix} | w |_{Ω_{T}}^{(2 + k + γ)} ⩽ C_{7} (| u_{0} |_{Ω}^{(3 + k + α)} + | f |_{Ω_{T}}^{(1 + k)} + | φ_{1} |_{Σ_{T}}^{(2 + k + α)}), \end{matrix}$ where $γ = 1$ is the Lipschitz index. When ε tends to 0, we shall have the set of the solutions of the problem (1.2)–(1.4) $\begin{matrix} {u_{ε}}, u_{ε} \in C_{x t}^{3 + k + α, \frac{3 + k + α}{2}} ({\overline{Ω}}_{T}), \end{matrix}$ and $u_{ε}$ satisfies an estimate (6.11)

The sequence ${u_{ε}}$ is bounded and compact in the space $C_{x t}^{2 + k + β, 1 + \frac{k + β}{2}} ({\overline{Ω}}_{T})$ , $β \in (0, α / 2)$ . We choose from the sequence ${u_{ε}}$ the converging subsequence in $C_{x t}^{2 + k + β, 1 + \frac{k + β}{2}} ({\overline{Ω}}_{T})$ $\begin{matrix} {u_{ε_{n}}}, n \to \infty, \end{matrix}$ and denote $\begin{matrix} lim_{n \to \infty} u_{ε_{n}} = : w (x, t) . \end{matrix}$

We shall have from here and due to an estimate (6.11) $\begin{matrix} | u_{ε_{n}} |_{Ω_{T}}^{(2 + k + β)} ⩽ C_{5} M_{3} + ε_{n} C_{6} M_{4}, \end{matrix}$ and $\begin{matrix} lim_{n \to \infty} | u_{ε_{n}} |_{Ω_{T}}^{(2 + k + β)} = | w |_{Ω_{T}}^{(2 + k + β)} ⩽ C_{5} M_{3} . \end{matrix}$

From the original set ${u_{ε}}$ we choose subset ${u_{ε_{n}}} \subset {u_{ε}}$ , $u_{ε} \in C_{x t}^{3 + k + α, \frac{3 + k + α}{2}} ({\overline{Ω}}_{T})$ , then ${\partial_{t} u_{ε_{n}}} \in C_{x t}^{1 + k + α, \frac{1 + k + α}{2}} ({\overline{Ω}}_{T})$ . The sequence ${\partial_{t} u_{ε_{n}}}$ is compact in $C_{x t}^{1 + k + β, \frac{1 + k + β}{2}} ({\overline{Ω}}_{T})$ with the same index $β \in (0, α / 2)$ as above thanks to the estimate $\begin{matrix} (6.12) & | \partial_{t} u_{ε_{n}} |_{Ω_{T}}^{(1 + k + α)} ⩽ C_{5} M_{3} + C_{6} ε_{n} M_{4} ⩽ C_{5} M_{3} + C_{6} ε_{0} M_{4} \end{matrix}$

Let $\begin{matrix} {\partial_{t} u_{ε_{n_{q}}}} \subset {\partial_{t} u_{ε_{n}}}, q \to \infty, \end{matrix}$ be subsequence converging in $C_{x t}^{1 + k + β, \frac{1 + k + β}{2}} ({\overline{Ω}}_{T})$ , we denote $\begin{matrix} lim_{q \to \infty} \partial_{t} u_{ε_{n_{q}}} (x, t) = : p (x, t), \end{matrix}$ then on the basis of (6.12) we obtain $\begin{matrix} | \partial_{t} u_{ε_{n_{q}}} |_{Ω_{T}}^{(1 + k + β)} ⩽ C_{5} M_{3} + C_{6} ε_{n_{q}} M_{4}, \end{matrix}$ and $\begin{matrix} lim_{q \to \infty} | \partial_{t} u_{ε_{n_{q}}} |_{Ω_{T}}^{(1 + k + β)} = | p |_{Ω_{T}}^{(1 + k + β)} ⩽ C_{5} M_{3} . \end{matrix}$

We have found the converging sequences ${u_{ε_{n}}}$ , it’s subsequence ${u_{ε_{n_{q}}}} \subset {u_{ε_{n}}}$ with the indexes $ε_{n_{q}}$ chosen above is also converging to $w (x, t)$ as a subsequence of the converging to $w (x, t)$ sequence ${u_{ε_{n}}}$ , i.e. $\begin{matrix} (6.13) & lim_{q \to \infty} u_{ε_{n_{q}}} = w (x, t), | w |_{Ω_{T}}^{(2 + k + β)} ⩽ C_{5} M_{3} . \end{matrix}$

Thus, we have constructed the sequences ${u_{ε_{n_{q}}}}$ and ${\partial_{t} u_{ε_{n_{q}}}}$ with the identical indexes converging as $q \to \infty$ to the functions $w (x, t)$ and $p (x, t)$ respectively. We point out that the function $p (x, t)$ is unknown, but exists.

We substitute the function $u_{ε_{n_{q}}}$ into the equation (1.2), initial and boundary conditions (1.3) (1.4) instead of u and $ε_{n_{q}}$ into (1.4) instead of ε, then letting q to ∞ we shall have that the function $w (x, t)$ is a solution of an unperturbed problem (1.5)–(1.7).

We increase the smoothness of $w (x, t)$ .

From the estimate (6.13) we shall have $\begin{matrix} (6.14) & | w |_{Ω_{T}}^{(2 + k)} ⩽ C_{5} M_{3} . \end{matrix}$

Consider the Hölder constants $\begin{array}{l} \sum_{2 m_{0} + | m | = 2 + k} ({[\partial_{t}^{m_{0}} \partial_{x}^{m} w]}_{x, Ω_{T}}^{(γ_{1})} + {[\partial_{t}^{m_{0}} \partial_{x}^{m} w]}_{t, Ω_{T}}^{(γ_{2})}) + \sum_{2 m_{0} + | m | = 1 + k} {[\partial_{t}^{m_{0}} \partial_{x}^{m} w]}_{t, Ω_{T}}^{(γ_{3})}, \\ (6.15) & γ_{j} \in (0, 1], j = 1, 2, 3, \end{array}$ where indexes $γ_{j}$ are unknown. We determine them.

We evaluate the first Hölder constant in (6.15). We have $\begin{array}{l} \frac{| \partial_{t}^{m_{0}} \partial_{x}^{m} w (x, t) - \partial_{t}^{m_{0}} \partial_{z}^{m} w (z, t) |}{| x - z |^{γ_{1}}} ⩽ & \frac{| \partial_{t}^{m_{0}} \partial_{x}^{m} w (x, t) - \partial_{t}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}} (x, t) |}{| x - z |^{γ_{1}}} \\ + \frac{| \partial_{t}^{m_{0}} \partial_{z}^{m} u_{ε_{n_{q}}} (z, t) - \partial_{t}^{m_{0}} \partial_{z}^{m} w (z, t) |}{| x - z |^{γ_{1}}} \\ (6.16) & + \frac{| \partial_{t}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}} (x, t) - \partial_{t}^{m_{0}} \partial_{z}^{m} u_{ε_{n_{q}}} (z, t) |}{| x - z |^{γ_{1}}}, \end{array}$ here $2 m_{0} + | m | = 2 + k$ .

As ${u_{ε_{n_{q}}}} \subset {u_{ε}}$ , $u_{ε} \in C_{x t}^{3 + k + α, \frac{3 + k + α}{2}} ({\overline{Ω}}_{T})$ we can take $γ_{1} = 1$ in the last ratio, make use of an estimate (6.11), then we shall have $\begin{matrix} (6.17) & \begin{matrix} [b] & \frac{| \partial_{t}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}} (x, t) - \partial_{t}^{m_{0}} \partial_{z}^{m} u_{ε_{n_{q}}} (z, t) |}{| x - z |^{γ_{1}}} \\ ⩽ C_{7} \sum_{i = 1}^{n} {| \partial_{x_{i}} (\partial_{t}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}}) |}_{Ω_{T}} ⩽ C_{8} M_{3} + C_{9} ε_{n_{q}} M_{4} . \end{matrix} \end{matrix}$

Further, on the basis of (6.13) $\begin{matrix} (6.18) & \partial_{t}^{m_{0}} \partial_{x}^{m} w (x, t) - \partial_{t}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}} (x, t) \to 0, \partial_{t}^{m_{0}} \partial_{z}^{m} u_{ε_{n_{q}}} (z, t) - \partial_{t}^{m_{0}} \partial_{z}^{m} w (z, t) \to 0 \end{matrix}$ as $q \to \infty$ in ${\overline{Ω}}_{T}$ .

We apply estimate (6.17) in the inequality (6.16), tend q to ∞ and taking into consideration (6.18) we obtain $\begin{matrix} (6.19) & \frac{| \partial_{t}^{m_{0}} \partial_{x}^{m} w (x, t) - \partial_{t}^{m_{0}} \partial_{z}^{m} w (z, t) |}{| x - z |^{γ_{1}}} ⩽ C_{10} M_{3}, γ_{1} = 1 . \end{matrix}$

We take supremum with respect to $(x, t), (z, t) \in {\overline{Ω}}_{T}$ in the inequality (6.19) and derive an estimate of the Hölder constant $\begin{matrix} {[\partial_{t}^{m_{0}} \partial_{x}^{m} u]}_{x, Ω_{T}}^{(γ_{1})} ⩽ C_{10} M_{3}, γ_{1} = 1 . \end{matrix}$

To estimate the second Hölder constant in (6.15) we write an inequality similar to (6.16) with $2 m_{0} + | m | = 2 + k$ $\begin{array}{l} \frac{| \partial_{t}^{m_{0}} \partial_{x}^{m} w (x, t) - \partial_{t_{1}}^{m_{0}} \partial_{x}^{m} w (x, t_{1}) |}{| t - t_{1} |^{γ_{2}}} ⩽ & \frac{| \partial_{t}^{m_{0}} \partial_{x}^{m} w (x, t) - \partial_{t}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}} (x, t) |}{| t - t_{1} |^{γ_{2}}} \\ + \frac{| \partial_{t_{1}}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}} (x, t_{1}) - \partial_{t_{1}}^{m_{0}} \partial_{x}^{m} w (x, t_{1}) |}{| t - t_{1} |^{γ_{2}}} \\ (6.20) & + \frac{| \partial_{t}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}} (x, t) - \partial_{t_{1}}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}} (x, t_{1}) |}{| t - t_{1} |^{γ_{2}}}, \end{array}$ apply the following estimate of the last ratio in (6.20), which is fulfilled for all $γ_{2} \in (0, \frac{1 + α}{2}]$ : $\begin{matrix} \frac{| \partial_{t}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}} (x, t) - \partial_{t_{1}}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}} (x, t_{1}) |}{| t - t_{1} |^{γ_{2}}} ⩽ C_{11} M_{3} + C_{12} ε_{n_{q}} M_{4}, \end{matrix}$ let $γ_{2} = γ_{1} / 2 = 1 / 2$ for the compatibility with $γ_{1} = 1$ , take into consideration the convergence of $\partial_{t}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}} (x, t)$ to $\partial_{t}^{m_{0}} \partial_{x}^{m} w (x, t)$ as $q \to \infty$ in ${\overline{Ω}}_{T}$ , then we shall have $\begin{matrix} {[\partial_{t}^{m_{0}} \partial_{x}^{m} w]}_{t, Ω_{T}}^{(γ_{1} / 2)} \equiv sup_{(x, t), (x, t_{1}) \in Ω_{T}} \frac{| \partial_{t}^{m_{0}} \partial_{x}^{m} w (x, t) - \partial_{t_{1}}^{m_{0}} \partial_{x}^{m} w (x, t_{1}) |}{| t - t_{1} |^{γ_{1} / 2}} ⩽ C_{11} M_{3} . \end{matrix}$

To find an estimate of the Hölder constants in the third sum in (6.15) with $2 m_{0} + | m | = 1 + k$ we consider an inequality $\begin{array}{l} \frac{| \partial_{t}^{m_{0}} \partial_{x}^{m} w (x, t) - \partial_{t_{1}}^{m_{0}} \partial_{x}^{m} w (x, t_{1}) |}{| t - t_{1} |^{γ_{3}}} ⩽ & \frac{| \partial_{t}^{m_{0}} \partial_{x}^{m} w (x, t) - \partial_{t}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}} (x, t) |}{| t - t_{1} |^{γ_{3}}} \\ + \frac{| \partial_{t_{1}}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}} (x, t_{1}) - \partial_{t_{1}}^{m_{0}} \partial_{x}^{m} w (x, t_{1}) |}{| t - t_{1} |^{γ_{3}}} \\ (6.21) & + \frac{| \partial_{t}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}} (x, t) - \partial_{t_{1}}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}} (x, t_{1}) |}{| t - t_{1} |^{γ_{3}}} . \end{array}$

Further, we have $\begin{matrix} (6.22) & \partial_{t}^{m_{0}} \partial_{x}^{m} w (x, t) - \partial_{t}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}} (x, t) \to 0 as q \to \infty \forall (x, t) \in {\overline{Ω}}_{T} . \end{matrix}$

In the last ratio in (6.21) we can assume $γ_{3} = 1$ , then $\begin{array}{l} \frac{| \partial_{t}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}} (x, t) - \partial_{t}^{m_{0}} \partial_{z}^{m} u_{ε_{n_{q}}} (x, t_{1}) |}{| t - t_{1} |^{γ_{3}}} \\ (6.23) & ⩽ C_{13} {| \partial_{t} (\partial_{t}^{m_{0}} \partial_{x}^{m} u_{ε_{n_{q}}}) |}_{Ω_{T}} ⩽ C_{14} M_{3} + C_{15} ε_{n_{q}} M_{4}, \end{array}$ because ${u_{ε_{n_{q}}}} \subset {u_{ε}}$ , $u_{ε} \in C_{x t}^{3 + k + α, \frac{3 + k + α}{2}} ({\overline{Ω}}_{T})$ and $2 m_{0} + | m | = 1 + k$ .

In the inequality (6.21) we make use of an estimate (6.23), tend $q \to \infty$ , then with the help of (6.22) we find $\begin{matrix} sup_{(x, t), (x, t_{1}) \in Ω_{T}} \frac{| \partial_{t}^{m_{0}} \partial_{x}^{m} w (x, t) - \partial_{t_{1}}^{m_{0}} \partial_{x}^{m} w (x, t_{1}) |}{| t - t_{1} |^{γ_{3}}} = {[\partial_{t}^{m_{0}} \partial_{t}^{m} u]}_{t, Ω_{T}}^{(γ_{3})} ⩽ C_{14} M_{3} . \end{matrix}$

Thus, the estimates (6.14) of $| w |_{Ω_{T}}^{(2 + k)}$ and of the Hölder constants lead to the following inequality $\begin{matrix} (6.24) & | w |_{Ω_{T}}^{(2 + k + γ)} ⩽ C_{16} (| u_{0} |_{Ω}^{(3 + k + α)} + | f |_{Ω_{T}}^{(1 + k + α)} + | φ_{1} |_{Σ_{T}}^{(2 + k + α)}), γ = 1 \end{matrix}$ and prove that the solution $w (x, t)$ of the unperturbed problem (1.5)–(1.7) belongs to the space $C_{x t}^{2 + k + γ, 1 + k / 2 + γ / 2} ({\overline{Ω}}_{T})$ , $γ = 1$ .

By the estimate (6.24) there is followed the uniqueness of the solution $w (x, t)$ of the linear unperturbed problem and convergence of the whole sequence ${u_{ε} (x, t)}$ to $w (x, t)$ as $ε \to 0$ .

We have lost the smoothness of the given functions, but thanks to this we have proved Theorem 3.5 without applying of Theorem 3.3. □

Footnotes

Acknowledgements

This work is supported by the Committee of Sciences of the Ministry of Education and Sciences of Republic of Kazakhstan, grant AP05133898.

Model problem

References

B.V.

Bazaliy, Stefan problem, Doklady AS USSR. Ser. A. 11 (1986), 3–7.

G.I.

Bizhanova, On the solvability of the Amann problem in the Hölder spaces, J. of Math. Sciences 127(2) (2005), 1828–1848. doi:10.1007/s10958-005-0145-8.

G.I.

Bizhanova, On the solutions of the linear free boundary problems of Stefan type, II, Mat. Zh., Almaty 12(2) (2012), 70–86.

G.I.

Bizhanova, Solvability of the nonregular problem for a parabolic equation with a time derivative in the boundary condition, Mat. Zh., Almaty 17(3) (2017), 62–70.

G.I.

Bizhanova, Convergence in the Holder space of the solutions to problems for parabolic equations with two small parameters in boundary condition, J. of Math. Sciences 236(4) (2019), 379–398. doi:10.1007/s10958-018-4119-z.

G.I.

Bizhanova, Solution of nonregular multidimensional two-phase problem for parabolic equations with time derivative in conjugation condition, Kazakh Math. J. 19(2) (2019), 31–48.

G.I.

Bizhanova and

M.N.

Shaimardanova, Solution of the nonregular problem for the heat equation with the time derivative in the boundary condition, Mat. Zh., Almaty 16(1) (2016), 35–57.

G.I.

Bizhanova and

V.A.

Solonnikov, On the solvability of an initial boundary-value problem for the parabolic equation with a time derivative in the boundary condition, St. Petersburg Math. J. 5(1) (1994), 97–124.

V.F.

Butuzov, On periodic solutions to singularly perturbed parabolic problems in the case of multiple roots of the degenerate equation, Comput. Math. and Math. Phys. 51(1) (2011), 40–50. doi:10.1134/S0965542511010064.

10.

V.F.

Butuzov, On asymptotics for the solution of a singularly perturbed parabolic problem with a multizone internal transition layer, Comput. Math. and Math. Phys. 58(6) (2018), 925–949. doi:10.1134/S0965542518060040.

11.

V.F.

Butuzov, Asymptotic behaviour of a boundary layer solution to a stationary partly dissipative system with a multiple root of the degenerate equation, Sbornik Math. 210(11) (2019), 1581–1608. doi:10.1070/SM9149.

12.

H.S.

Carslaw and

J.C.

Jaeger, Conductions of Heat in Solid, 2nd edn, Oxford, 1959.

13.

Chipot, On some anisotropic singular perturbation problems, Asymptotic Analysis 55(3–4) (2007), 125–144.

14.

Chipot, Asymptotic Issues for Some Partial Differential Equations, Imperial College Press, 2016.

15.

Chipot and

F.J.

Correa, Boundary layer solutions to functional elliptic equations, Bulletin of the Brazilian Mathematical Society, New Series 40 (2009), 381–393. doi:10.1007/s00574-009-0017-9.

16.

Chipot,

Guesmia and

Sengouga, Singular perturbations of some nonlinear problems, Journal of Mathematical Sciences, New York 176(6) (2011), 828–843. doi:10.1007/s10958-011-0439-y.

17.

A.R.

Danilin, Asymptotics of a solution to a problem of optimal boundary control with two small cosubordinate parameters, Trudy Inst. Mat. i Mekh. UrO RAN 26(1) (2020), 102–111. doi:10.21538/0134-4889-2020-26-1-102-111.

18.

G–M.

Gie,

Hamouda,

C.Y.

Jung and

Temam, Singular Perturbations and Boundary Layers, Applied Mathematical Sciences, Springer, 2018.

19.

G.-M.

Gie,

Ch.-Y.

Jung and

Temam, Recent progresses in boundary layer theory, Discrete and Continuous Dynamical Systems 36(5) (2016), 2521–2583. doi:10.3934/dcds.2016.36.2521.

20.

E.I.

Hanzawa, Classical solutions of the Stefan problem, Tohoku Math. J. 33(3) (1981), 297–335. doi:10.2748/tmj/1178229399.

21.

M.I.

Imanaliyev, in: Asymptotic Methods in the Theory of Singularly Perturbed Integro-Differential Systems, Ilim, Frunze, 1972.

22.

Ch.-Y.

Jung,

Park and

Temam, Boundary layer analysis of nonlinear reaction–diffusion equations in a smooth domain, Advances in Nonlinear Analysis 6(3) (2017), 277–300. doi:10.1515/anona-2015-0148.

23.

Ch.-Y.

Jung and

Temam, Boundary layer theory for convection–diffusion equations in a circle, Uspekhi Mat. Nauk 69(3) (2014), 43–86. doi:10.4213/rm9584.

24.

K.A.

Kasymov, Singularly perturbed boundary–value problems with initial jumps, Sanat, Almaty, 1997.

25.

O.A.

Ladyzhenskaja,

V.A.

Solonnikov and

N.N.

Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, “Nauka”, Moscow, 1967.

26.

A.M.

Meirmanov, The Stefan Problem, Walter de Gruyter & Co., Berlin, 1992.

27.

A.V.

Nesterov, The asymptotic form of the solution of an elliptic equation with the singularly perturbed boundary condition, Comput. Math. and Math. Phys. 34(11) (1994), 1477–1481.

28.

O.A.

Oleinik, On equations of elliptic type with a small parameter at the highest derivatives, Mat. Sbornik 31(1) (1952), 104–117.

29.

E.V.

Radkevich, On the solvability of the general nonstationary free boundary problems, in: Some Applications of Functional Analysis to the Problems of Mathematical Physics, Novosibirsk, 1986, pp. 85–111.

30.

V.A.

Solonnikov, On an estimate of the maximum of a derivative modulus for a solution of a uniform parabolic initial boundary–value problem, 1977, LOMI, Preprint No. P-2-77.

31.

Temam and

Wang, Asymptotic analysis of Oseen type equations in a channel at small viscosity, Indiana University Mathematics Journal 45(3) (1996), 863–916. doi:10.1512/iumj.1996.45.1290.

32.

S.I.

Temirbulatov, The problem of heat conduction with the time derivative in the boundary condition, Differentcial’nye uravneniya 19(4) (1983), 85–111.

33.

A.N.

Tikhonov, Systems of differential equations containing a small parameter multiplying the derivative, Mat. sbornik 31 (1952), 575–586.

34.

A.B.

Vasil’eva, On certain alternating boundary–layer solutions to singularly perturbed parabolic equations, Comput. Math. and Math. Phys. 48(4) (2008), 618–626. doi:10.1134/S096554250804009X.

35.

M.I.

Vishik and

L.A.

Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameter, Uspekhi Mat. Nauk 12(5) (1957), 3–122.