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Confluence of the nonlinear waves in the Stefan problem with undercooling

We assume that the Stefan problem with undercooling has a classical solution until the moment of contact of free boundaries and the free boundaries have continuous velocities until the moment of contact. Under these assumptions, we construct a smooth approximation of the global solution of the Stefan problem with undercooling, which, until the contact, gives the classical solution mentioned above and, after the contact, gives a solution which is the solution of the heat equation.


Introduction
In this paper, we study the confluence of free boundaries in the Stefan problem with kinetic undercooling in the spherical symmetric case.
We shall construct a smooth approximation of solutions of the Stefan problem with undercooling under the assumption that the motion of the free boundary is the motion of the front of a nonlinear wave and the confluence of free boundaries is interaction of solitary nonlinear waves. The possibility of this interpretation is given by the models of phase field [2] proposed by G. Caginalp. In fact, the choice of the method for approximating the limit Stefan problems with undercooling is unessential for us, because we do not prove that the approximations thus constructed are close to the corresponding solutions of the phase field system.
For example, we could use the definition of the generalized solution of the limit problems including the order function (the nonlinear wave) [10].
Here our considerations are based on the following simple fact. Suppose that there are two families of solutions (exact and approximate solutions) of some problem depending on a small parameter ε. Suppose that both these families have the properties that permit passing to the limit as ε → 0 in the weak sense. Suppose also that the family of approximate solutions satisfies a problem with a right-hand side small as ε → 0 in the weak sense. Then the weak limits of both families are solutions of the same limit problem and if the latter has a unique solution, then the difference between the exact and approximate solutions tends to 0 as ε → 0 at least in the weak sense.
We recall that the phase field system has the form where The function θ = θ(r, t, ε) has the meaning of the temperature, and the function u = u(r, t, ε), which is called the order function, determined the phase state of the medium: u ≃ −1 corresponds to the phase "−" in the layer Ω − (t), and u ≃ 1 corresponds to the phase "+" in the layers Ω − 1,2 (t). Passing to the limit as ε → 0 in (1), we obtain the Stefan problem with undercooling.
This passage to the limit is possible, for example, in the case where the corresponding limit problems have classical solutions. In this case, the weak limits as ε → 0 of solutions (1) give these solutions [2,5,13].
We denote θ(r, t) = σ(r, t) r . ( Then the phase field system (1) we can write in the form The smooth approximations of solutions of the Stefan problem with kinetic undercooling constructed in this paper are approximate (in the above sense) solutions of the phase field system and they admit a weak passage to the limit as ε → 0. In this case, we obtain the limit problems (and their solutions) describing the process of confluence of free boundaries. To construct these approximations, we use the assumption that the classical sharp fronted solution of the Stefan problem with kinetic undercooling exists until the confluence of free boundaries begins. This is a natural assumption in our paper (otherwise, it is not clear the confluence of what is considered) and can be proved 1 .
We repeat that, under this assumption, we construct an approximation of solution of the limit problem, which is smooth for ε > 0 and uniformly bounded in ε 0. As is well known, in a similar problem about the propagation of shock waves, the existence of such an approximation distinguishes a unique solution.
By t * ∈ [0, t 1 ) we denote the instant of confluence of free boundaries. Then, for any t t * − δ for all δ > 0, we see that the limit problems have solutions. The asymptotic solution of system (3),(4) has the form Here ). By S(R n ) we denote the Schwartz space of smooth rapidly decreasing functions. If the initial data for (1) has the form (5), (6) at t = 0, then, for t t * − δ, we have the estimate where (σ, u) is a solution of system (3),(4) (see [1,5]). Here Q = Ω×[0, t * −δ), and the constant c is independent of ε.
The main obstacle to the construction of solutions of the form (5), (6), which could be used to describe the confluence of free boundaries, is the fact that, instead of an ordinary differential equation whose solution is the function ω 0 (z) [2,5], in the case of confluence of free boundaries, we must deal with a partial differential equation for which the explicit form of the exact solution is unknown.
In the present paper, we use the technique of the weak asymptotics method [4,8], which allows us to avoid this problem. Let us explain several basic points.
where the constant C t 1 ,ζ depends on t 1 and the test function ζ(r).
Generalizing (7), we shall say that the family of distributions f (r, t, ε) depending on t and ε as on parameters admits the estimate O D ′ (ε ν ) if, for any test function ζ(r), we have the estimate f (r, t, ε), ζ(x) = O(ε ν ), 0 t t 1 .
Example 4. The following relation is a corollary of Definition 1: for all m ∈ Z + and α > 0. Generally speaking, this is not true for the derivatives w.r.t. t.
The relations in Examples 2 (a), 2 (b) and 4 are obvious, but, for the reader's convenience, we prove some of these formulas in Section 5.
Here we only note that, in view of items (a) and (b), one can say that the product of approximations of the Heaviside functions or of the Heaviside functions themselves is a linear combination with accuracy up to small terms (∼ O D ′ (ε)). The general nonlinear functions of linear combinations of approximations of the Heaviside functions can be linearized similarly. Precisely this property underlies the constructive study of the interaction between nonlinear waves with localized fast variations. Definition 2. A pair of smooth functions (ǔ,σ) is a weak asymptotic solution of the phase field system (3),(4) if, for any test functions ζ(r), ξ(r) ∈ C ∞ 0 (Ω), the following relations hold: The left-hand side of Eq. (9) is obtained by multiplying equation (4) byǔ r and integrating by parts. The reminders O(ε α ), α = 1, µ, in the right-hand sides of (8) and (9) must be bounded locally in t, i.e., for t ∈ [0, t 1 ], we have This construction was introduced and analyzed in [6]. The fact that a noninteger exponent appears in the right-hand side of (9) is not directly related to the technique of the weak asymptotics method, see Examples 1-4. The source of this noninteger exponent is the nonsmoothness of the function θ (the temperature), which appears at the instant of confluence of the free boundaries.
The paper is organized as follows.
In Section 2, we explain the structure of ansatzes of approximations of the temperature and the order function. These ansatzes are constructed under the assumption the classical sharp fronted solution of the Stefan problem exists. At the beginning of Section 2, we formulate what we exactly need.
Next, in Section 3, we substitute the constructed ansatzes in system (3),(4) and derive equations for the unknown functions contained in the ansatzes.
The results of these calculations are summarized in Theorems 1-3 in Section 3. The final result of this paper is formulated in Theorem 4 in Section 3. It states that the assumption on the existence of the classical solution is sufficient for constructing formulas for the weak asymptotic solution of system (3),(4) in the sense of Definition 1. Next, in Section 4, we analyze the constructed formulas and derive the following effects: (a) the weak asymptotic solution is smooth for t > t * , the absolute values of the free boundaries velocities are equal to each other at the contact moment; (b) the temperature has a negative jump at the instant and at the point of confluence of the free boundaries, and this jump is equal to where r = r i0 (t) other positions of the free boundaries till the interaction.
In particular, it follows from (a) and (b) that the velocities of the free boundaries have jumps at the point of contact.
These effects can also be discovered in the numerical analysis of the process of confluence of free boundaries. Some helpful technical results are given in Section 5. We note that the only example known to the authors, where the confluence of the free boundaries is studied, is given in [11].
In general, this paper turned out to be technically rather complicated and long in spite of the fact that the details in several justifications in Section 5 were omitted.
Although, as was shown above, the results obtained here are a necessary step in the study of the multidimensional problem, this paper shows that the following classification can be introduced: (1) problems in which the confluence of free boundaries leads to disappearance of one of the phases; (2) problems in which the domain occupied by one of the phases changes its connectivity, but the number of phases remains the same.
In this paper, we consider an example precisely from the first class of problems. As was mentioned above, we do not justify the asymptotics of the constructed solution. But this can be done based on our constructions. It is easy to see that the main role here is played by the (not proved) existence of the classical solution up to the moment of confluence of the free boundaries. Our construction of the existnece under this assumption reduced the justification to estimating the soluton of the heat equation with the right-hand side f ε admitting the estimate µ ∈ (0, 1/2) and with zero initial and boundary conditions. An analysis of the structure of this right-hand side shows that f ε · ε −µ as ε → 0 is a linear combination of functions δ ′ (r − r i ) and δ(r − r i ), i = 1, 2 with coefficients depending on t, τ , and these coefficients converge fast to zero as τ → ±∞.
Hence we can conclude that for r = r i the solution of this heat equation belongs to C ∞ for ε ≥ 0 and admits the estimate O(ε µ ). In the whole domain Ω × [0, T ], a rough analysis based on general theorems [14] shows that the solution belongs to W −δ 2 , δ > 0, and has the estimate O(ε µ ) in the norm of this space. In particular, the weak limit of the constructed weak asymptotic solution is equal to the exact global solution of the heat equation in the phase field system.

Ansatz of the approximation of the solution of the Stefan problem with undercooling
In our case, the Stefan problem with kinetic undercooling has the form Relations (10)- (12) are supplemented with (Dirichlet or Neumann) boundary conditions for r = R i , i = 1, 2, and with a consistent initial condition. Obviously, problem (10)- (12) can be written as We shall assume that the initial conditions are chosen so that the problem in question has the classical solution,r 2 (0) −r 1 (0) > 0, and there exists a . More precisely, we assume that (hereD denotes the interior of the domain D), cf. [3].
As will be shown below, these assumptions permit describing the interaction (the confluence of free boundaries) constructively.
We shall construct this approximation (ansatz) as a weak asymptotic solution of system (3),(4).
First, we introduce the ansatz of the order function in (10)- (12). It has the formǔ Here the unknowns are the functions β and In what follows, we write them more precisely. Now we only note that if β > 0, then, for r 1 < r 2 , the functionǔ coincides up to O(ε N ) with the sum of the first three terms in the right-hand side of (14), for r 1 = r 2 , we haveǔ = 1+ O D ′ (ε), and for r 1 > r 2 , we haveǔ = 1 + O(ε N ). These relations can easily be verified directly. We note that, under the above assumptions, we can continue the functionsr i (t) to the interval [0, t 1 ] preserving the smoothness and the sign of the derivatives. We choose such continuations and denote them by r i0 (t). Now, we can write the functions r i (t, ε) and β(t, ε) more precisely. Namely, we set where We note that as ε → 0. Thus, the value of the variable τ characterizes the process of confluence of free boundaries. Also, we introduce the functions In addition, we assume that Assumptions (19) for the functions r i1 are the assumption that as ε → 0 the functions r i (t, ε) approximate continuous functions with a possible discontinuity (jump) of the derivatives at t = t * . More precisely, the functions r i (t, ε) determined in (15) are approximations of the functions Now we describe the ansatz of the approximation of the temperaturē σ(r, t). Here we also must use the continuation procedure for matching the temperature for t < t * , when there are three subdomains of the domain Q, and the temperature for t > t * , when there is only one phase. For this, we first introduce a "model" of the temperature whose continuation is reduced to the problem of continuation of functions depending only on the time t. This model must have the same structure as the temperatureσ. Obviously, the simplest function of this form is a function linear in x in the layers Ω + l,r (t) and quadratic in the layer Ω − (t).
Namely, we set The functions γ ± j = γ ± j (t), j = 1, 2, depend on t, are continuous for t < t * , and have limits as t → t * − 0. Therefore, they can be continued to the interval [0, t 1 ], t 1 > t * , so that the properties of being continuous and the signs are preserved. We shall use the previous notation for the continued functions. Now we can introduce the temperature modeľ where γ − (r, t),γ(r, t), and I(r, t) are linear functions of r, γ − | r=r i = γ − i (t), γ| r=r i =γ i (t), and I| r=r i = (−1) i+1 (r i r it + κ 2 ). In more detail, the functionŝ γ(r, t) and γ − (r, t) can be written in the form We have the relations This implies that, for t < t * , We rewrite the product as follows Here In what follows, we write B(τ ) instead of B(ρ). The difference is that B(ρ) is an unknonwn function (ρ is unknown), while B(τ ) is a known function.
But we do not use this remark and, by definition, we seť Now we can write the expression ∂ 2Ť /∂r 2 uniformly in ψ = r 2 − r 1 . We have We set γ + 1 = −γ 1 and γ − 2 =γ 2 and see that the coefficients for δ(x − r i ) in (24) take the form We note that the replacement of B(ρ) by B(τ ) implies a correction of order O D ′ (ε). Now we note that the inequalities hold for ψ = 0. Also, we note that ∂Ť /∂r is a smooth function. So, by (22) and the Stefan condition (12) we have for t < t * and assume that the continuations of the functions contained in these relations are chosen so that these relations hold for t ∈ [0, t 1 ].
The assumption that σ(r, t) is the classical solution of problem (10) and the limitq exists as t → t * − 0 and for x ∈ Ω. Therefore, we can continue the functionq to the domain Ω × [0, t 1 ], where t 1 > t * and the properties of smoothness are preserved, and moreover, We shall construct the global temperatureθ in the form whereq is the desired function, e(r)

Construction of the weak asymptotic solution
We first consider the heat equation (3). Using the formulas of the weak asymptotics method and taking (14) into account, we obtain We assume that functionq and its continuation are chosen so that the functionq satisfies the boundary conditions of the original problem. Then the boundary conditions for the functionq are zero.
We substitute the functionσ determined by relation (27) and expression (28) for ∂ǔ/∂t into Eq. (3). We denotê According to the definition 2 with accuracy O D ′ (ε), we obtain We let τ tend to ∞ (i.e., for t < t * ) and, in view of (10) and (18), obtain Since eŤ +q → σ as τ → ∞, we see thatq → 0 as τ → ∞. The total equation for the function q =q +q has the form where F (r, t, τ ) is a piecewise continuous function containing of terms that are uniformly bounded in ε on Ω × [0, t 1 ]. Equation (29) was derived under the assumption that Byq i ,q * i we denote the terms in q such that With accuracy up to functions smooth in Ω, we can calculate them using the fundamental solution of the heat equation (in what follows, we shall consider only the first equation), It is clear that the functionsq i are uniformly bounded. It is also clear that . Therefore, to justify Eq. (29), we must verify that no δ-functions arise in calculating the derivatives ∂ 2q i /∂r 2 , ∂ 2q * i /∂r 2 and ∂q i /∂t, ∂q * i /∂t. For this, we note that, for any test function ζ(r) up to functions smooth in Ω, we have the relation It is clear that f (ξ, t) ∈ C ∞ (Ω) for all t. Calculating the derivative ∂q 1 /∂r, we obtain cannot hold for any coefficients G i . Similarly to (33), with accuracy up to smooth functions, we can write the functionsq * i , i = 1, 2. They have the same properties asq i with the only additional condition This easily follows from the explicit formula of the type of (31) and the fact that, for r = r * , the integral with respect to ξ is an integral of an odd function over a symmetric interval. Therefore, condition (32) is necessary for deriving Eq. (29). This implies that the functionq + eI is uniformly bounded in Ω × [0, t 1 ] and belongs to C 1,0 for r = r i , i = 1, 2.
So if the functionq + eI satisfies Eq. (29) with zero initial conditions and the boundary conditions that follow from the fact that the functionθ = eŤ +q +q must satisfy the boundary conditions of the original problem, while the functionsŤ andq are known (more precisely, they will be determined after the functions ϕ i are found), then we have the following assertion.
Theorem 1. Suppose that the functionσ is determined by relation (27), and the functionq is a solution of Eq. (29) with zero initial condition and zero boundary conditions. Suppose that relation (30) holds.
Then the pair of functionsσ andǔ is a weak asymptotic solution of the heat equation in the phase field system, i.e., relation (3) holds. Remark 1. Expression (30) may seem to be strange, because we did not take into account the well-known fact that the Dirac functions are linearly independent. But this fact is taken into account in the framework of the weak asymptotics method (see Lemma 2, Section 5). Here we do not want to mix the substitution of the ansatz into the equation and the analysis of the results of this substitution (see Section 4). Now we consider the Allen-Cahn equation (4). We must calculate the weak asymptotics of the expression where the functionσ is determined by the relation (27). We obtain where V i j , i, j = 1, 2, are linear combinations of several convolutions. Their expressions will be given below. Here we note the following. We have We denote We have the following estimates: where |f (z, η)| dz ≤ const, |f 1 (z, η)| dz ≤ const.
These relations readily follow from the explicit form of the function ω 0 (z), In fact, relations (37) and (38) express the above-described properties of the ansatzǔ (14) in different terms.
Using the technique of the weak asymptotics method (see Lemma 7 in Section 5), we can write Similarly, we obtain (see Lemmas 5 and 7, Section 5): Thus, adding expressions (35), (39) and (40), we see that the coefficients of the δ-functions in formula (34) have the form where Similarly, we obtain (see Lemma 6, Section 5): Thus, to obtain β = β(η) we have the equation (obviously, this is a necessary condition for the relation F = O D ′ (ε) to hold, see (34)): According to (43) and (45) (D is positive), the right-hand side of the relation (46) is positive. So we have proved the following assertion.
Theorem 2. Suppose that the assumptions of Theorem 1 and (46) are satisfied and Then the pair of functionsσ,ǔ is a weak asymptotic solution of the phase field system (3), (4) in the sense of Definition 2.
Thus, under the assumption that the classical solution of the phase field system exists (see (i) and (ii) above), relations (29), (30), (46), and (47) are sufficient conditions for constructing a weak asymptotic solution of system (3), (4). In what follows, we prove that the relations mentioned above are equations for determining the functions β = β(τ ) and r i1 (τ ), i = 1, 2. We present an algorithm for solving these equations.

An analysis of the relations obtained
We begin with relation (30) and, for a while, forget everything said about the notion of linear independence.
Then, for this relation to hold, it suffices to have the two relations In view of (22), we have γ + i + γ − i = ∂σ ∂r r=r i (t) . Taking this into account and adding relations (48) and Lemma 2, we obtain Here we used the relation which follows from the definition of the function ρ, see (17). We set the last term to be zero, since we show below that r 21 − r 11 = O(|τ | −1 ). In view of Lemma 3 in Section 5, in this case we have (r 21 − r 11 ) t ψ 0 = O(ε). Moreover, in view of the Stefan conditions and the choice of the continuation of the functions γ ± i and r 0i (t), we have We transform the left-hand side of relation (49): Next, we change the variables in the last integral Finally, we get Since the function ω 0 (z) is odd, we see that the expression in braces in the first integral in the right-hand side is Hence, integrating by parts, we obtain Or, finally, So, in view of (49), we obtain the equation for ρ: Or ∂ ∂τ η + η tanh η β = 2B(τ ).
Since B(τ ) = O(|τ | −N ) for any N as τ → −∞, the integral in the righthand of (52) converges and hence the left-hand side tends to the limit Moreover, in view of the inequality B 0, we have η 0 for τ ∈ R 2 τ , which implies that η → 0 as τ → −∞.
Next, we have Moreover, as readily follows from the exponential rate of convergence to the limit of the functionsĈ/D, 1 − B, and 1 + tanh η as τ → ±∞, the derivatives satisfy the estimate

Now we calculate the limits of the expressions
Therefore, we have lim in view of the Stefan conditions (12), and J i = O(τ −1 ), i = 1, 2.
Proof. In view of the corollary of Lemma 2, Section 5, the estimate for the coefficients of the δ-functions in (30) as τ → ∞ and (54) are sufficient conditions for relation (30) to follow from (49). In turn, relation (53) follows from (49).
Now we analyze the relations V 2 i = 0 and (47). We use the explicit form of the function F (z) and the formula As η → ∞, we have and hence β → 1 as η → ∞.
From relation (36) we easily obtain Similarly to (56), using (48), we can verify the relations We rewrite the expressions for the coefficients V 1 i in more detail: We take the limit in formulas (58), (59) as τ → ∞ i.e. prior to interaction. So, in view of conditions (43), (57) we obtain σ r r=r 10 (t) =Ĉ + r ′ 10 (t) +Ĉ whereĈ + = ω 2 0 (z)dz. In view of (11) we obtain κ 1 = κ 2 =Ĉ + . Thus, in view of the corollary of Lemma 2, Section 5, for the relation Thus, we have proved the following assertion. We consider relation (60). It follows from the above that C Ω = C Ω (η) decreases sufficiently fast as η → ∞, in any case ||η|C Ω (η)| const. This fact, the estimatê proved in Lemma 4 in Section 5, relation (54) implies that the obvious estimate for t t * . For t t * , by Lemma 6, we have where ψ 1 = r 21 − r 11 , Hence, by (17), and a statement similar to Lemma 3, in Section 5, we have 2(r 10t + r 20t ) − q r r=r 10 + q r r=r 20 +Ĉ + 1 r 10 We introduce a function V (τ ) ∈ C ∞ , such that V ′ τ ∈ S(R 1 ), V (−∞) = 0, V (∞) = 1. Then, in view of considerations similar to those used in Lemma 3, we can show that the following estimate hold: This implies that the left-hand side of (34) is estimated as From this relation we obtain From this equation we determine the extensions of the functions r 11 and r 21 . We note that our argument results in an equation that does not contain the temperature, namely, the assumption that the classical solution exists till t = t * is sufficient for constructing a global smooth approximation of the solution.
Let us calculate the function B Ω = B Ω (η) in more detail. We have In this integral, we make the change of variable z → −z − η. Then we obtain Hence we have B Ω (η) = Ω ′ z Ω ′ η dz.
Finally, we obtain Thus, for finite η, the denominator (B Ω + C Ω ) in the last term in the righthand side of (62) does not vanish. Moreover, using the explicit form of the functions ω 0 (z), B z Ω , B Ω , and C Ω , we can verify that where N ≫ 1 is an arbitrary number. Therefore, we have We note that where ρ = ηβ −1 and the functions η and β are determined by Eqs. (53) and (46). The system of Eqs. (63) and (64) allows one to find the functions r i1 , i = 1, 2, and thus completely determines the functions contained in the ansatz of the weak asymptotic solution of system (3), (4).
It is easy to see that the solutions thus constructed satisfy conditions (19). Next, using the explicit formulas for B z Ω , B Ω , and C Ω , we can easily verify that B z Ω | η=0 = 0 (because ω 0 (z) is odd) and (B Ω + C Ω )| η=0 = 0, becauseω 0 (z) is even. This means that We note that the right-hand side of (29) is a piecewise smooth function for |ψ| const > 0 and smooth for r = r i , i = 1, 2. As ψ → 0 (t → t * ), the righthand side of (29) becomes proportional to δ(r−r * 0 ), where r * 0 = r * | ψ=0 , and the proportionality coefficient is negative. Thus, if r and t vary in the respective neighborhoods of the points r * 0 and t * , then the δ-function with a negative coefficient appears and disappears in the right-hand side of (29), which results in a negative soliton-like jump of the temperature in a neighborhood of the point t = t * , r = r * 0 . Calculation of the temperature jump. To prove the statement in the Introduction concerning the temperature jump, we must calculate the quantity [eI +q +q]| t=t * , r=r * .
Let us verify thatq +q is a continuous function. This function is the sum of solutions of the heat equation with singular right-hand sides, but these singularities arise after the substitution of the continuous function eŤ (which, of course, is not a solution) into the left-hand side. It remains to prove that the solution differs from the function eŤ by a continuous function. For this, it suffices to verify that the right-hand sides, which arise after the substitution of eŤ into the equation, do not generate any singularities in the solution of the heat equation in addition to those contained in the function eŤ .
Here we, in contrast to the preceding statements, use the fact that the singularities in the right-hand sides arise as the result of the substitution.
Byq i , i = 1, 2, we denote the terms inq+q corresponding to the right-hand sides: The other terms in q are solutions of the heat equation with piecewise continuous right-hand side and hence are continuous. The functionsq i have a similar property if they are calculated up to O(ε). Indeed, we denote Π = γ − (r 1 − r)(r − r 2 ) ψ and represent, for example, the functionq 1 in the form Omitting the number factor and integrating by parts in the integral over ξ, we obtain The last term is a solution of the heat equation with the piecewise continuous right-hand side and hence it is continuous uniformly w.r.t ε ≥ 0. The other terms are also continous functions uniformly w.r.t ε ≥ 0. It is easy to see that in the formula forq 1 there is no term containing the derivative ∂B ∂t , because this term would be of order O(ε).
Let us calculate the function eI. We have It follows from (51) thatρ So, the jump of the temperature has the form

Technique of the weak asymptotics method
First, we recall the definition of regularization of the generalized function.

Definition 3.
A family of functions f (x, ε) smooth for ε > 0 and satisfying the condition is called the regularization of the generalized functionf (x).
We note that, by definition, the last relation can be rewritten as for any test function ζ(x) (from now on, , denotes the action of a generalized function on a test function).
where ϕ(t) is a smooth function, let ω(z) ∈ S (S is the Schwartz space), and let β = β(t) > 0. Then the following relation holds for any test function ζ(x): The expression in the right-hand side can be written as Here we perform the change of variables z = β(x − ϕ)/ε and apply the Taylor formula to the integrand at the point x = ϕ. By definition, the last integral is the action of the generalized function β −1 A ω δ(x − ϕ) on the test function ζ.
Proof of the formula in Example 1 in the Introduction. Let ω(z) ∈ C ∞ , ω ′ ∈ S(R 1 ), lim z→+∞ ω(z) = 0, and lim z→−∞ ω(z) = 1. We verify that where H is the Heaviside function. By Definition 1, we consider the expression Here we performed the change of variables z = (x − x 0 )/ε and applied the Taylor formula to the functions ζ(x) at the point x = x 0 . In view of our assumptions and the properties of the function H, the integral in the righthand side of the last relation converges and hence the right-hand side of (67) is of order O(ε). We thus obtain estimate (66).
Proof of the formula in Example 2 in the Introduction. Let ω i (z) ∈ C ∞ , (ω i ) ′ ∈ S(R 1 ), lim z→+∞ ω i (z) = 0, and lim z→−∞ ω i (z) = 1, i = 1, 2. We consider the integral Integrating by parts, we obtain We perform the change of variables z = (x − x i )/ε with i = 1 in the first integral and with i = 2 in the second integral and apply the Taylor formula to the function F (x) = x −∞ ζ(y) dy at the points x = x i , i = 1, 2, respectively. Then we calculate the first and the second integral, we have To calculate the linear combination of generalized functions up to O D ′ (ε α ), we must improve the classical definition of linear independence. This improvement plays the key role in the study of problems with interaction of nonlinear waves.
Indeed, let φ 1 = φ 2 be independent of x. We consider the expression where the functions g i are independent of ε. Clearly, for the last relation to be satisfied, it suffices to have or, with the properties of the functions g i taken into account, But, if we assume that the coefficients g i depend on the parameter ε, then the above estimates do not work. Namely, let us consider the following specific case of this dependence: where the functions S i (σ) decrease sufficiently fast as |σ| → ∞, and △φ = φ 2 − φ 1 .
If S 1 = −S 2 then expression (68) can be rewritten as Thus we obtain the assertion of Lemma 2.
Proof. Following the above argument, we obtain In view of our assumptions, uniformly in t, which implies the desired assertion.
Proof. The fraction f (t) t−t 0 is locally bounded in t. The fraction τ g(t,τ ) t−t 0 is also locally bounded. We have According to the assumptions of the lemma, the last factor in the right-hand side is bounded on the interval under study. The product of the first and second factors (without ε) is bounded in view of the properties of the function g(t, τ ).

Corollary 2.
Suppose that the assumptions of Lemma 3 are satisfied for 0 τ < ∞ (−∞ < τ 0). Then the assertion of Lemma 3 holds on any half-interval (t 0 ,t], which does not contain zeros of the function f (t), and signt = signf (t), t ∈ (t 0 ,t]. The proof of Corollary 2 is obvious. Proof. We consider only one of the the functionsq from (32), namely, the functions determined by relation (33), and apply the method developed in [14]. We consider the difference of expressions (32) omitting the number and apply the estimate which, obviously, is valid in the required range of variables. Hence, as in Lemma 4, we obtain the estimate J = O(ε µ ), µ ∈ (0, 1/2), which proves relation (71).
Similarly to Lemmas 4 and 5, we prove the following statement.
Lemma 6. The following relation holds for any of the functionsq i defined by relations (31):q Then the left-hand side of (77) is positive, and hence the integrand expression in the right-hand side is also positive (until the measure of the points at which |u r | const is positive).
To prove this relation, we must successively perform the change of variables (75) in the integral in the left-hand side and then consider the half-sum of the integrals obtained, which, obviously, is equal to the original integral in the left-hand side. Moreover, we must take into account the estimates (37), (38), and Ω(z, η) = 1 + O(exp(−2z)), z → ∞, ω 0 (−z − η) + O(exp(2z)), z → −∞, which together with the explicit form of the function F (u) = u 4 4 − u 2 2 + 1