GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 6, 1998, 501-512

ASYMPTOTIC EXPANSION OF SOLUTIONS OF PARABOLIC EQUATIONS WITH A SMALL PARAMETER

A. GAGNIDZE

Abstract. The heat equation with a small parameter,

(1 + ε^{−m}χ( \frac{x}{ε}))u_{t} = u_{xx}, is considered, where ε ∈ (0, 1), m < 1 and χ is a finite function. A complete asymptotic expansion of the solution in powers ε is constructed.

In [1], [2] E. Sanchez-Palencia and H. Tchatat noted, for the first time, problems in which a small parameter is contained not only in the equation but also in the characteristics of the domain itself. In subsequent years such problems were studied by O. A. Oleynik, S. A. Nazarov, Yu. D. Golovatii, and G. S. Sobolev [3]–[5].

In this paper we consider a problem on heat conduction in a medium whose density has a perturbation concentrated in a small neighborhood of the origin.

In the domain Ω = (1, 1) × (0, T) let us consider the initial boundary value problem for a heat equation of the form

(1 + ε^{−m}χ(\frac{x}{ε}))\frac{∂u}{∂t} = \frac{∂^{2}u}{∂x^{2}}  (1)

with the boundary conditions

u(−1, t) = u(1, t) = 0  (2)

and the initial condition

u(x, 0) = u_{0}(x),  (3)

where ε ∈ (0, 1), m < 1 is some real number, and the function χ satisfies the following conditions: χ(ξ) = 0 for |ξ| > 1, χ(ξ) > 0 for |ξ| < 1, and

501 ∫_{−1}^{1} χ(ξ)dξ = M = const > 0. Assume that the initial function u_{0} is continu-

1991 Mathematics Subject Classification. 35C20.

Key words and phrases. Heat equation, small parameter, concentrated perturbation, complete asymptotic expansion.

1072-947X/98/1100-0501$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

ous on [−1, 1], satisfies the condition u_{0}(1) = u_{0}(1) = 0, and is holomorphic in the neigborhood of x = 0.

In that case it readily follows that

lim_{ε→0} ε^{−m} ∫_{−ε}^{ε} χ(\frac{x}{ε}) dx = 0

and therefore such a perturbation (m < 1) will be called “weak”.

By a solution of problem (1)–(3) we shall understand a function u which satisfies equation (1) in Ω for x ≠ ±ε, conditions (2) and (3), and at the discontinuity points x = ±ε of the function χ (there are no other discontinuity points) satisfies the conditions of “continuous sewing”

\begin{matrix}u(ε + 0, t) = u(ε − 0, t), & \frac{∂u}{∂x}(ε + 0, t) = \frac{∂u}{∂u}∂x(ε − 0, t), \\ u(−ε + 0, t) = u(−ε − 0, t), & \frac{∂u}{∂x}(−ε + 0, t) = \frac{∂u}{∂u}∂x(−ε − 0, t). \end{matrix} (4)

According to O. A. Oleynik’s paper [6] problem (1)–(3) is uniquely solvable in the domain Ω.

Let m be a rational number and m = \frac{l}{p}, where p is a natural number and ℓ < p is some integer number. We introduce the notation ξ = \frac{x}{ε}, Ω_{+}^{ε} = (ε, 1) × (0, T), Ω_{−}^{ε} = (−1, −ε) × (0, T), Ω_{+} = (0, 1) × (0, T), Ω_{−} = (−1, 0) × (0, T).

Now we shall construct a complete asymptotic expansion of the solution u_{ε} of problem (1)–(3) in powers of value δ = ε^{\frac{1}{p}} when ε → 0.

We shall seek for a solution of the form

u_{ε}(x, t) ∼ \begin{cases}∑_{i=0}^{∞} δ^{i}v_{i}^{±}(x, t), & (x, t) ∈ Ω_{±}^{ε}, \\ ∑_{i=0}^{∞} δ^{i}w_{i}(\frac{x}{ε}, t), & (x, t) ∈ (−ε, ε) × (0, T). \end{cases}  (5)

First we shall find out which conditions the functions v_{i}^{±} and w_{i} must satisfy when t = 0.

By virtue of expansion (5) and condition (3) we have

u_{0}(x) ∼ ∑_{i=0}^{∞} δ^{i}v_{i}^{±}(x, 0), |x| > ε.

Hence it follows that

v_{0}^{±}(x, 0) = u_{0}(x), v_{i}^{±}(x, 0) = 0, i ≥ 1.  (6)

Using expansion (5) and condition (3), for the function w_{i} we obtain

u_{0}(x) ∼ ∑_{i=0}^{∞} δ^{i}w_{i}(ξ, 0), |ξ| < 1,

which, after expanding u_{0} into a Taylor series, gives

∑_{i=0}^{∞} δ^{i}w_{i}(ξ, 0) ∼ ∑_{i=0}^{∞} δ^{pi} \frac{ξ^{i}}{i!} \frac{d^{i}}{dx^{i}} u_{0}(0).

Having equated the coefficients at the same powers of δ, we obtain

\begin{matrix}w_{ip}(ξ, 0) = \frac{ξ^{i}}{i!} \frac{d^{i}}{dx^{i}} u_{0}(0), & i = 0, 1, 2, …, \\ w_{j}(ξ, 0) = 0, j ≠ pk, & k = 0, 1, 2, …. \end{matrix}  (7)

It is easy to verify that condition (2) implies

v_{i}^{±}(±1, t) = 0, i ≥ 0.  (8)

By substituting the formal expansion (5) into equation (1) we obtain

∑_{i=0}^{∞} δ^{i}(\frac{∂}{∂t} v_{i}^{±}(x, t) − \frac{∂^{2}}{∂x^{2}} v_{i}^{±}(x, t)) ∼ 0,

∑_{i=0}^{∞} δ^{i−2p}(\frac{∂}{∂t} w_{i−2p}(ξ, t) + χ(ξ) \frac{∂}{∂t} w_{i−2p+ℓ}(ξ, t) − \frac{∂^{2}}{∂ξ^{2}} w_{i}(ξ, t)) ∼ 0,

which yields

\frac{∂}{∂t}v_{i}^{±}(x, t) − \frac{∂^{2}}{∂x^{2}} v_{i}^{±}(x, t) = 0, |x| > ε,  (9)
\frac{∂^{2}}{∂ξ^{2}} w_{i}(ξ, t) = \frac{∂}{∂t}w_{i−2p}(ξ, t) + χ(ξ) \frac{∂}{∂t}w_{i−2p+ℓ}(ξ, t), |ξ| < 1, (10)

where there are no terms with negative indices.

In what follows we shall write v_{i} instead of v_{i}^{±}, assuming that for x > ε we mean v_{i}^{+}, and for x < −ε we mean v_{i}^{−}. Let, in the neighborhood of the point (0, 1), the functions v_{i} be holomorphic with respect to x.

By the formal expansion (5) we obtain

u_{ε}(x, t) ∼ ∑_{i=0}^{∞} δ^{i} ∑_{s=0}^{∞} \frac{x^{s}}{s!} \frac{∂^{s}}{∂x^{s}} v_{i}(±0, t), |x| > ε,

\frac{∂u_{ε}}{∂x} (x, t) ∼ ∑_{i=0}^{∞} δ^{i} ∑_{s=1}^{∞} \frac{x^{s−1}}{(s − 1)!} \frac{∂^{s}}{∂x^{s}} v_{i}(±0, t), |x| > ε,

u_{ε}(x, t) ∼ ∑_{i=0}^{∞} δ^{i}w_{i}(ξ, t), |ξ| < 1,

\frac{∂u_{ε}}{∂x} (x, t) ∼ ∑_{i=0}^{∞} δ^{i−p} \frac{∂w_{i}}{∂ξ} (ξ, t), |ξ| < 1.

But the solution u_{ε} must satisfy the condition of “continuous sewing” (4). Therefore for x = ε and ξ = 1 we have

\begin{matrix}∑_{i=0}^{∞} δ^{i} ∑_{s=0}^{[ \frac{i}{p}]} \frac{1}{s!} \frac{∂^{s}}{∂x^{s}} v_{i−ps}(+0, t) ∼ ∑_{i=0}^{∞} δ^{i}w_{i}(1, t), \\ ∑_{i=0}^{∞} δ^{i} ∑_{s=0}^{[ \frac{i}{p}]} \frac{1}{s!} \frac{∂^{s+1}}{∂x^{s+1}} v_{i−ps}(+0, t) ∼ ∑_{i=0}^{∞} δ^{i} \frac{∂w_{i+p}}{∂ξ} (1, t). \end{matrix}  (11)

Hence

w_{i}(1, t) = ∑_{s=0}^{[ \frac{i}{p}]} \frac{1}{s!} \frac{∂^{s}}{∂x^{s}} v_{i−ps}(+0, t), i ≥ 0.

Therefore

w_{i}(1, t) − v_{i}(+0, t) = ∑_{s=1}^{[ \frac{i}{p}]} \frac{1}{s!} \frac{∂^{s}}{∂x^{s}} v_{i−ps}(+0, t).

Thus we obtain

w_{i}(1, t) − v_{i}(+0, t) = F_{i}^{+}, i ≥ 0,  (12)

where the value F_{i}^{+} is defined by the values of v_{j}(+0, t) for j ≤ i − p. In the same manner we obtain

w_{i}(−1, t) − v_{i}(−0, t) = F_{i}^{−}, i ≥ 0,  (13)

where F_{i}^{−} is defined by the values of v_{j}(−0, t) for j ≤ i − p.

For the functions w_{i} we obtain

\frac{∂w_{i}}{∂ξ} (1, t) = 0, i = 0, 1, …, (p − 1),

\frac{∂w_{i}}{∂ξ} (1, t) = ∑_{s=0}^{[ \frac{i}{p}]−1} \frac{1}{s!} \frac{∂^{s+1}}{∂x^{s+1}} v_{i−p−ps}(+0, t), i ≥ p.

As a result, we have

\frac{∂w_{i}}{∂ξ} (1, t) − \frac{∂}{∂x} v_{i−p}(+0, t) = Φ_{i}^{+},  (14)

where Φ_{i}^{+} depend on v_{j} for j ≤ i − 2p.

In the same manner we obtain

\frac{∂w_{i}}{∂ξ} (−1, t) − \frac{∂}{∂x} v_{i−p}(−0, t) = Φ_{i}^{−},  (15)

where Φ_{i}^{−} depends on v_{j} for j ≤ i − 2p.

It will be shown now how one can construct successively all the functions v_{i} and w_{i}.

I. Step 1. By equation (10) we have \frac{∂^{2}w_{i}}{∂ξ^{2}} (ξ, t) = 0, i = 0, 1, …, (p − 1). Then \frac{∂w_{i}}{∂ξ} (ξ, t) = a_{i}(t), i = 0, 1, …, (p − 1). But condition (14) implies \frac{∂w_{i}}{∂ξ} (±1, t) = 0, i=0, 1, …, (p−1). Then \frac{∂w_{i}}{∂ξ} (ξ, t) = 0 for i = 0, 1, …, (p−1). Therefore w_{i}(ξ, t) = C_{i}(t), i = 0, 1, …, (p−1). By equation (10) we obtain \frac{∂^{2}w_{p}}{∂ξ^{2}} (ξ, t) = 0 and \frac{∂w_{p}}{∂ξ} (ξ, t) = a_{p}(t). But (14) implies that \frac{∂w_{p}}{∂ξ} (±1, t) = \frac{∂v_{0}}{∂x} (±0, t). Then for the function v_{0} we obtain the condition \frac{∂v_{0}}{∂x} (+0, t) = \frac{∂v_{0}}{∂x} (−0, t). Condition (12) obviously implies v_{0}(+0, t) = v_{0}(−0, t). Thus to define the function v_{0} we obtain the problem

\frac{∂v_{0}}{∂t} (x, t) = \frac{∂^{2}v_{0}}{∂x^{2}} (x, t), x ≠ 0,

v_{0}(x, 0) = u_{0}(x),

v_{0}(−1, t) = v_{0}(1, t) = 0, v_{0}(+0, t) = v_{0}(−0, t),

\frac{∂v_{0}}{∂x} (+0, t) = \frac{∂v_{0}}{∂x} (−0, t),

which, as follows from [6], is uniquely solvable. Moreover, the solution coincides with the solution of the problem

\frac{∂v_{0}}{∂t} (x, t) = \frac{∂^{2}v_{0}}{∂x^{2}} (x, t), x ≠ 0,

v_{0}(−1, t) = v_{0}(1, t) = 0, v_{0}(x, 0) = u_{0}(x).

Thus the function v_{0} is defined uniquely. But in that case the condition w_{0}(±1, t) = v_{0}(±0, t) implies w_{0}(ξ, t) = C_{0}(t) = v_{0}(0, t).

Therefore, by performing step 1, we uniquely define the functions v_{0} and w_{0}, while the functions w_{1}, w_{2}, …, w_{p} are defined to within the functions C_{i} depending only on t.

II. Step 2. By equation (10) we obtain

\frac{∂^{2}w_{p+1}}{∂ξ^{2}} (ξ, t) = χ(ξ) \frac{∂w_{1+ℓ−p}}{∂t} (ξ, t),

where ℓ < p and the right-hand part is absent if 1 + ℓ − p < 0.

Thus for w_{p+1} we obtain the equation

\frac{∂^{2}w_{p+1}}{∂ξ^{2}} (ξ, t) = f_{0}(ξ, t),

where f_{0} is the known function. Hence it follows that

\frac{∂w_{p+1}}{∂ξ} (ξ, t) = ∫_{ξ_{0}}^{ξ} f_{0}(s, t) ds + a_{p+1}(t)

and \frac{∂w_{p+1}}{∂ξ} is defined to within a term of the form a_{p+1}(t). In step 1 we have defined w_{1} to within the term C_{1}(t). By condition (12) we have

w_{1}(±1, t) − v_{1}(±0, t) = 0 if p > 1;

w_{1}(±1, t) − v_{1}(±0, t) = \frac{∂}{∂x} v_{0}(±0, t) if p = 1.

In both cases it is easy to verify that v_{1}(+0, t) − v_{1}(−0, t) = h_{1}(t), where h_{1} is uniquely defined.

By condition (14) we have

\frac{∂w_{p+1}}{∂ξ} (±1, t) = \frac{∂v_{1}}{∂x} (±0, t) if p > 1;

\frac{∂w_{p+1}}{∂ξ} (±1, t) = \frac{∂v_{1}}{∂x} (±0, t) + \frac{∂^{2}v_{0}}{∂x^{2}} (±0, t) if p = 1.

In both cases this readily yields

\frac{∂v_{1}}{∂x} (+0, t) − \frac{∂v_{1}}{∂x} (−0, t) = \frac{∂w_{p+1}}{∂ξ} (+1, t) − \frac{∂w_{p+1}}{∂ξ} (−1, t) + h̃_{0}(t),

where h̃_{0} depends on v_{0}. Therefore

\frac{∂v_{1}}{∂x} (+0, t) − \frac{∂v_{1}}{∂x} (−0, t) = H_{1}(t),

where H_{1} is uniquely defined.

Thus to define the function v_{1} we obtain the problem

\frac{∂v_{1}}{∂t} (x, t) = \frac{∂^{2}v_{1}}{∂x^{2}} (x, t), x ≠ 0,

v_{1}(x, 0) = 0,

v_{1}(−1, t) = v_{1}(1, t) = 0, v_{1}(+0, t) − v_{1}(−0, t) = h_{1}(t),

\frac{∂v_{1}}{∂x} (+0, t) − \frac{∂v_{1}}{∂x} (−0, t) = H_{1}(t),

where h_{1} and H_{1} are the known functions. This problem is uniquely solvable according to [6].

Thus the function v_{1} is defined uniquely. Since the function w_{1}(ξ, t) = C_{1}(t), by the condition w_{1}(1, t) − v_{1}(+0, t) = 0 for p > 1 and the condition w_{1}(1, t) − v_{1}(−0, t) = \frac{∂}{∂x}v_{0}(+0, t) for p = 1 the function w_{1} is also defined uniquely.

The function w_{p+1} can be represented by

\frac{∂w_{p+1}}{∂ξ} (ξ, t) = ∫_{ξ_{0}}^{ξ} f_{0}(s, t) ds + a_{p+1}(t).

Then, by the conditions

\frac{∂w_{p+1}}{∂ξ} (1, t) = \frac{∂v_{1}}{∂x} (+0, t) for p > 1,

\frac{∂w_{p+1}}{∂ξ} (1, t) = \frac{∂v_{1}}{∂x} (+0, t) + \frac{∂^{2}v_{0}}{∂x^{2}} (+0, t) for p = 1,

the function a_{p+1} is uniquely defined. Therefore the function \frac{∂w_{p+1}}{∂ξ} is

uniquely defined and \frac{∂w_{p+1}}{∂ξ} (ξ, t) = f_{1}(ξ, t). Hence we obtain

w_{p+1}(ξ, t) = ∫_{ξ_{0}}^{ξ} f_{1}(s, t) ds + C_{p+1}(t)

and the function w_{p+1} is defined to within the term C_{p+1} depending on t.

Thus in step 2 we have defined the functions w_{1} and v_{1} uniquely, while the function w_{p+1} was defined to within the function C_{p+1} depending on t.

III. Step n + 1. Let the functions v_{i} and w_{i} be uniquely defined for all i ≤ n, and the functions w_{n+1}, …, w_{n+p} be defined to within the terms C_{n+1}, …, C_{n+p} depending on t.

Consider the equation for the function w_{n+p+1}

\frac{∂^{2}w_{n+p+1}}{∂ξ^{2}} (ξ, t) = \frac{∂w_{n+1−p}}{∂t} (ξ, t) + χ(ξ) \frac{∂w_{n+1+ℓ−p}}{∂t} (ξ, t),

where the right-hand part has no terms with negative indices. In any case the right-hand part of the equation if defined uniquely since ℓ < p and p ≥ 1, and therefore n + 1 − p ≤ n and n + 1 + ℓ − p ≤ n. As a result we obtain the equation

\frac{∂^{2}w_{n+p+1}}{∂ξ^{2}} (ξ, t) = f_{n}(ξ, t)

which readily implies that

\frac{∂w_{n+p+1}}{∂ξ} (ξ, t) = ∫_{ξ_{0}}^{ξ} f_{n}(s, t) ds + a_{n+p+1}(t)

and the function \frac{∂w_{n+p+1}}{∂ξ} is defined to within the term a_{n+p+1} depending only on t.

By condition (12) we have w_{n+1}(±1, t) − v_{n+1}(±0, t) = F_{n+1}^{±}, where F_{n+1}^{±} is defined by means of the functions v_{j} for j ≤ n + 1 − p. But n+1−p ≤ n and therefore the difference v_{n+1}(+0, t)−v_{n+1}(−0, t) is defined by the difference w_{n+1}(1, t) − w_{n+1}(−1, t), which is uniquely defined. Thus v_{n+1}(+0, t) − v_{n+1}(−0, t) = h_{n+1}(t), where h_{n+1} is the known function.

By condition (14) we have

\frac{∂w_{n+1+p}}{∂ξ} (±1, t) − \frac{∂v_{n+1}}{∂x} (±0, t) = Φ_{n+p+1}^{±}(t),

where Φ_{n+p+1}^{±} is defined by means of the functions v_{j} for j ≤ n + 1 − p. But n + 1 − p ≤ n and therefore the difference \frac{∂}{∂x}v_{n+1}(+0, t) − \frac{∂}{∂x}v_{n+1}(−0, t) is defined by the difference

\frac{∂w_{n+p+1}}{∂ξ} (1, t) − \frac{∂w_{n+p+1}}{∂ξ} (−1, t) = ∫_{−1}^{1} f_{n}(s, t) ds

and hence is defined uniquely. Thus

\frac{∂v_{n+1}}{∂x} (+0, t) − \frac{∂v_{n+1}}{∂x} (−0, t) = H_{n+1}(t),

where H_{n+1} is the known value.

Finally, to define the function v_{n+1} we obtain the problem

\frac{∂v_{n+1}}{∂t} (x, t) = \frac{∂^{2}v_{n+1}}{∂x^{2}} (x, t), x ≠ 0,

v_{n+1}(x, 0) = 0,

v_{n+1}(−1, t) = v_{n+1}(1, t) = 0,

v_{n+1}(+0, t) − v_{n+1}(−0, t) = h_{n+1}(t),

\frac{∂v_{n+1}}{∂x} (+0, t) − \frac{∂v_{n+1}}{∂x} (−0, t) = H_{n+1}(t),

where h_{n+1} and H_{n+1} are the known functions. This problem is uniquely solvable according to [6].

Thus the function v_{n+1} has been defined uniquely. Since the function w_{n+1} has been defined to within the term C_{n+1} depending only on t, the function w_{n+1} is defined uniquely by the condition w_{n+1}(1, t) − v_{n+1}(+0, t) = F_{n+1}^{+}(t).

The function w_{n+p+1} can be represented as

\frac{∂w_{n+p+1}}{∂ξ} (ξ, t) = ∫_{ξ_{0}}^{ξ} f_{n}(s, t) ds + a_{n+p+1}(t).

Then the function a_{n+p+1} is defined uniquely by the condition

\frac{∂w_{n+p+1}}{∂ξ} (1, t) − \frac{∂v_{n+1}}{∂x} = Φ_{n+p+1}^{+}(t).

Therefore

\frac{∂w_{n+p+1}}{∂ξ} (ξ, t) = f_{n+1}(ξ, t)

and w_{n+p+1} is defined by the formula

w_{n+p+1}(ξ, t) = ∫_{ξ_{0}}^{ξ} f_{n+1}(s, t) ds + C_{n+p+1}(t)

to within the term C_{n+p+1}.

Therefore, if it is assumed that the functions v_{i} and w_{i} have the known exact values for all i ≤ n, and the fucntions w_{n+1}, …, w_{n+p} are known to within the terms C_{n+1}, …, C_{n+p} depending only on the variable t, then we shall define the functions v_{n+1} and w_{n+1} uniquely, while the function w_{n+1+p} will be defined to within the term C_{n+p+1} depending only on t.

Thus, using the arguments of I, II, and III, we conclude by induction that the functions v_{i} and w_{i} can be defined uniquely for arbitrary i. We have therefore formally constructed the asymptotic series (5).

Consider a partial sum of series (5)

u_{N} (x, t) = \begin{cases}∑_{i=0}^{N} δ^{i}v_{i}^{±}(x, t), & |x| > ε, \\ ∑_{i=0}^{N} δ^{i}w_{i}(\frac{x}{ε}, t), & |x| < ε, \end{cases}  (16)

and evaluate the difference u_{N} (ε + 0, t) − u_{N} (ε − 0, t).

We readily obtain

u_{N} (ε − 0, t) = ∑_{i=0}^{N} δ^{i}w_{i}(1, t);

u_{N} (ε + 0, t) = ∑_{i=0}^{N} δ^{i} ∑_{s=0}^{[ \frac{i}{p}]} \frac{1}{s!} \frac{∂^{s}}{∂x^{s}} v_{i−sp}(+0, t) + O(δ^{N+1}).

But then conditions (11) imply that u_{N} (ε + 0, t) − u_{N} (ε − 0, t) = O(δ^{N+1}). In the same manner we find that

u_{N} (−ε + 0, t) − u_{N} (−ε − 0, t) = O(δ^{N+1});
\frac{∂u_{N}}{∂x} (ε + 0, t) − \frac{∂u_{N}}{∂x} (ε − 0, t) = O(δ^{N});
\frac{∂u_{N}}{∂x} (−ε + 0, t) − \frac{∂u_{N}}{∂x} (−ε − 0, t) = O(δ^{N}).

Thus the function u_{N} has discontinuities at the points x = ±ε. Let us correct this function at the points of discontinuity. We introduce the notation

C_{1} = u_{N} (ε + 0, t) − u_{N} (ε − 0, t),
C_{2} = u_{N} (−ε + 0, t) − u_{N} (−ε − 0, t),
B_{1} = \frac{∂u_{N}}{∂x} (ε + 0, t) − \frac{∂u_{N}}{∂x} (ε − 0, t),
B_{2} = \frac{∂u_{N}}{∂x} (−ε + 0, t) − \frac{∂u_{N}}{∂x} (−ε − 0, t).

Let φ be a smooth function, φ(x) ≡ 1 for |x| ≤ \frac{1}{2}, and φ(−1) = φ(1) = 0. Assume that φ_{N} (x, t) for |x| < ε, φ_{N} (x, t) = (B_{1}(x−ε)+C_{1})φ(x) for x ≥ ε, and φ_{N} (x, t) = (B_{2}(x + ε) + C_{2})φ(x) for x < −ε. Consider the function V_{N} defined by the formula V_{N} (x, t) = u_{N} (x, t) − φ_{N} (x, t). It is easy to obtain

(\frac{∂V_{N}}{∂t} + ε^{−m}χ(\frac{x}{ε}) \frac{∂V_{N}}{∂x^{2}} ) = O(δ^{N−1}),

V_{N} (x, 0) = u_{0}(x) + O(δ^{N+1}).

Now for the function \overline{v} = u_{ε} − V_{N} we obtain a problem of the form

\begin{matrix}\frac{∂\overline{v}}{∂t} + ε^{−m}χ(\frac{x}{ε}) \frac{∂\overline{v}}{∂t} − \frac{∂^{2}\overline{v}}{∂x^{2}} = F_{δ}, \\ \overline{v}(−1, t) = \overline{v}(1, t) = 0, \\ \overline{v}(x, 0) = φ_{δ}, \end{matrix}  (17)

where F_{δ}(x, t) = O(δ^{N−1}) and φ_{δ}(x) = O(δ^{N+1}).

By multiplying the equation by \overline{v} and integrating the resulting equality over the domain [−1, 1] × [0, τ_{0}], where τ_{0} ∈ (0, T], we obtain

∫_{−1}^{1} ∫_{0}^{τ_{0}}((1 + ε^{−m}χ(\frac{x}{ε}))\frac{∂\overline{v}}{∂t} \overline{v}^{2} − \frac{∂^{2}\overline{v}}{∂x^{2}} \overline{v})dx dt = ∫_{−1}^{1} ∫_{0}^{τ_{0}}F_{δ}(x, t)\overline{v} dx dt.

Hence, after integration by parts, we have

\frac{1}{2} ∫_{−1}^{1} (1 + ε^{−m}χ(\frac{x}{ε}))\overline{v}^{2} dx + ∫_{−1}^{1} ∫_{0}^{τ_{0}}(\frac{∂\overline{v}}{∂x})^{2}dx dt =

= \frac{1}{2} ∫_{−1}^{1} (1 + ε^{−m}χ(\frac{x}{ε}))φ_{δ}^{2}(x) dx + ∫_{−1}^{1} ∫_{0}^{τ_{0}}F_{δ}(x, t)\overline{v}(x, t) dx dt,

which implies

∫_{−1}^{1} \overline{v}^{2}(x, τ_{0}) dx ≤ ∫_{−1}^{1} φ_{δ}^{2}(x) dx + ε^{−m} ∫_{−ε}^{ε} χ(\frac{x}{ε})φ_{δ}^{2}(x) dx +

+ 2| ∫_{−1}^{1} ∫_{0}^{τ_{0}}F_{δ}(x, t)\overline{v}(x, t) dx dt|.

Taking into account the estimates of the functions φ_{δ} and F_{φ} and using the known inequality 2ab ≤ εa^{2} + \frac{1}{ε}b^{2}, we obtain

∫_{−1}^{1} ∫_{0}^{T} \overline{v}^{2}(x, t) dx dt ≤ C̃δ^{2(N−1)},

where C̃ does not depend on δ and N.

Thus we have established that ∥u_{ε} − V_{N_{1}} ∥_{L_{2}(Ω)} ≤ C̃δ^{N_{1}−1} for any N_{1}. Let N_{1} = N + 2. Then ∥u_{ε} + V_{N+2}∥_{L_{2}(Ω)} ≤ C̃δ^{N+1}. On the other hand,

∥V_{N+2} − u_{N+2}∥_{L_{2}(Ω)} ≤ \overline{C}δ^{N+2}. Hence it follows that ∥u_{ε} − u_{N+2}∥_{L_{2}(Ω)} ≤ C_{1}δ^{N+1}. This immediately implies ∥u_{ε} −u_{N} ∥_{L_{2}(Ω)} ≤ M̃δ^{N+1}. Thus we have proved

Theorem. Let u_{ε} be the solution of problem (1)–(3) and u_{N} be a partial sum of the formal asymptotic series (5) defined by formula (16). Then the inequality ∥u_{ε} − u_{N} ∥_{L_{2}(Ω)} ≤ M̃δ^{N+1} holds, where the constant M̃ does not depend on δ and N.

The construction of the functions v_{j} and w_{j} enables us to make several conclusions. In particular, let m_{1} = \frac{ℓ_{1}}{p} and m_{2} = \frac{ℓ_{2}}{p}, where ℓ_{1} < ℓ_{2}. It is easy to see that in both cases the functions v_{0} and w_{0} are defined in the same manner. Moreover, the functions v_{j} and w_{j} are also defined in the same manner if j < p − 1 − ℓ_{2}. Thus for such m_{1} and m_{2} the asymptotic expansions coincide in the first several terms.

The theorem and the above remarks give rise to

Corollary. Let m < 1 be a real number. Then the limit function \overline{u} = lim_{ε→0} u_{ε}, where u_{ε} is the solution of problem (1)–(3), is the solution of the problem

\frac{∂\overline{u}}{∂t}(x, t) = \frac{∂^{2}\overline{u}}{∂x^{2}}(x, t), x ∈ (−1, 1),

\overline{u}(−1, t) = \overline{u}(1, t) = 0, \overline{u}(x, 0) = u_{0}(x).

Remark. The corollary can also be proved without using asymptotic expansions. We intend to do this in future papers.

References

1. E. Sanchez-Palencia, Perturbation of eigenvalues in thermoelasticity and vibration of systems with concentrated masses. Lecture Notes in Phys., No. 155, 316–368, Springer-Verlag, Berlin, 1984.

2. E. Sanchez-Palencia and H. Tchatat, Vibration de systemes elastiques avec des masses concentrees. Rend. Sem. Mat. Univ. Politec. Torino 42(1984) No. 3, 43–63.

3. Y. D. Golovatii, S. A. Nazarov, O. A. Oleynik, and T. S. Sobolev. On eigenoscillations of a string with the adjoint mass. (Russian) Sibirsk. Mat. Zh. 29(1988), No. 5, 71–91.

4. Yu. D. Golovatii, S. A. Nazarov, and O. A. Oleynik, The asymptotics of eigenvalues and eigenfunctions in problems on oscillation of a string with singular perturbation of the density. (Russian). Uspekhi Mat. Nauk 43(1990), No. 5, 189–190.

5. Yu. D. Golovatii, S. A. Nazarov, and O. A. Oleynik, Asymptotic expansion of eigenvalues and eigenfunctions of problems on oscillation of a medium with concentrated perturbations. (Russian) Trudy Mat. Inst. Steklov. 192(1990), 42–60.

6. O. A. Oleynik, Boundary value problems for linear equations of elliptic and parabolic type with discontinuous coefficients. (Russian) Izv. Akad. Nauk SSSR, Ser. Mat. 25(1961), No. 1, 3–20.

(Received 30.04.1996)

Author’s address:

Faculty of Mechanics and Mathematics

I. Javakhishvili Tbilisi State University

2, University St., Tbilisi 380043

Georgia