On a Boundary Value Problem for a Quasi-linear Elliptic Equation Degenerating into a Parabolic Equation in an Infinite Strip

Аbstract In this paper we construct complete asymptotics on the small parameter of the solution of a singularly perturbed boundary value problem for a quasi-linear elliptic equation degenerating in an infinite strip into the parabolic equation, and the remainder term is estimated.


Introduction
While studying numerous real phenomena with non-uniform transitions from one physical characteristics to another ones, we have to investigate singularly perturbed boundary value problems.A lot of papers have been devoted to the asymptotics of the solution of different boundary value problems for nonlinear elliptic equations with a small parameter at higher derivatives.In a great number of papers on nonlinear singularly perturbed elliptic equations, the input equations degenerate for a zero value of the small parameter into functional equations (see.[1]- [4], [6]).Besides, in all these papers with the exception of the paper [4], the derivatives of the desired function enter linearly to the equation, only the desired function itself enters into the equation nonlinearly.All these and other problems known to us are considered only in finite domains.
In the present paper, in an infinite strip we consider the following boundary value problem , 0 ) , (

The first iterative process
In the first iterative process, we'll look for the approximate solution of equation (1) in the form and the functions ) y , x ( W i will be chosen so that ( ) (4) Substituting ( 3) in ( 4), expanding the nonlinear terms in powers of ε and equating the terms with the same powers of ε , for determining the function n i W i ,..., 1 , 0 ; = we get the following recurrently connected equations: where , and the functions s g are dependent polynomially on the first and second derivatives of . We'll solve equations (5) under the following boundary conditions: , problem (5), ( 6) is said to be a degenerated problem corresponding to problem (1), (2).
The following lemma is valid Lemma 1.Let ) , ( y x f be a function given in Π , having continuous derivatives with respect to x to the ( ) -th order inclusively, be infinitely differentiable with respect to y and satisfy the condition

3
where l is a nonnegative number, is arbitrary, . Then the function ( ) being the solution of problem ( 5), (6) , in Π has continuous derivatives with respect to x to the ( 2 )-th order inclusively, is infinitely differentiable with respect to y and satisfies the condition where Proof.Applying the Fourier transformation with respect to y , problem (5), (6) for Here . The solution of problem (9) is of the form is found as the inverse Fourier transformation of the function ) , x ( W ~λ 0 from the following formula: From condition (7) it follows that the function ) , ( ~λ x f and all its derivatives with respect to x to the ( 1 + n )-th order inclusively, with respect to the variable λ belong to the S L.Schwarts space (in the sequel we'll denote it by λ S ).Obviously, for proving lemma 1 it suffices to show that the function that is the solution of problem (9), and all its derivatives with respect to x to the ( ) -th order inclusively belong to λ S .By the mathematical induction method we can prove the validity of the following formula: Here ( ) is a polynomial with respect to λ and τ − x , more exactly, ( ) , moreover the coefficients r C are real numbers and some of them may equal zero.
. While obtaining (14) we used . It is easy to show that the derivatives of the function with respect to x of any order are expressed by the formula The functions with respect to λ have a polynomial growth.Above we proved the belongness of the function to the space λ S .Thus each summand contained in the right hand side of (15) is the product of two functions, one of them has a polynomial growth, the another one enters into the space λ S .Therefore, the relation is valid, whence it follows that ( ) 1 contained in expansion (3) will be sequentially determined from boundary value problems ( 5), (6) for n i ,..., 2 , 1 = .From lemma 1 it follows that the functions i W being the solutions of problems (5), (6) for n i ,..., 2 , 1 = , will have continuous derivatives with respect to x to the th order and condition (8) for the function i W will be satisfied for From ( 3) and ( 6) we get that the constructed function W satisfies the following boundary conditions: The function W doesn't satisfy, generally speaking, boundary condition (2) for 1 = x . For compensating the missed boundary condition it is necessary to construct a boundary layer type function near the boundary 1 = x .

The Second Iterative Process-Construction of Boundary Layer Functions
where j Φ are the known functions dependent on ...
) as the solution of the equation Expanding each function ( ) Substituting expressions (18), (21) for the functions W V , to (19) and taking into account (17), for determining 1 1 0 ,..., , we get the following equations: where j Q are the known functions dependent on , ,..., , , ,..., , their first and second derivatives.We can write the formulae for j Q obviously, but they are of bulky form.Here we give formulae only for 1 Q and The boundary conditions for equations ( 22), (23) are obtained from the requirement that the sum V W + should satisfy the boundary condition 0 Substituting the expressions for W and V , respectively from ( 3) and ( 18) into (24), taking into account that we look for 1 ,..., 1 , 0 ; as a boundary layer type function, we have ( ) where ( ) ( ) The following lemma is valid ) has a unique solution that is infinitely differentiable with respect to both variables τ and y .And the following estimation is valid BVP for quasi-linear elliptic equation where ) ,..., , ( are some known polynomials of their own arguments with non-negative coefficients, the free members of these polynomials equal zero, and even one of other coefficients is non-zero. Proof.Existence and uniqueness of the solution of problem ( 22), ( 25) for 0 = j were proved in [5] (see theorem 2).The solution of problem ( 22), ( 25) for 0 = j in the parametric form is as follows ( ) where t is a parameter, ( ) , then the corresponding real root ) ( 0 0 y t of algebraic equation ( 28) also vanishes and the expression for τ in with respect to τ was also proved in [5].But there ) ( 0 y ϕ has continuous derivatives with respect to y to definite finite order.In connection with the fact that here y S ) y ( ∈ ϕ 0 , is infinitely differentiable, ( ) y t 0 also will be an infinitely differentiable function.Hence it follows an infinite differentiability of ( ) with respect to y .
Prove the validity of estimation (26).From the first equality of (27), we can get an estimation of the form Having transformed equation (28) we have: [ ] . Hence it is seen that the function . Consequently, from (29) we get the following estimation Taking into account (30), in the second equality of (27) we have ( ) Recalling that the parametric form of the solution of problem ( 22), (25) for 0 = j was obtained by means of substitution q V = ∂ ∂ τ 0 , from (30) we get an We can represent the function where ( ) denotes the following function: ( ) ( ) , from (32), (33) we get an estimation for with respect to τ , we differentiate sequentially the both parts of (33) with respect to τ , and each time take into account the estimations of previous derivatives.These estimations will be of the form (32), i.e. ( ) For 0 = j , from (25) we get that the function ψ should satisfy the boundary conditions 0 lim ), ( The solution of problem (35), (36) is of the form BVP for quasi-linear elliptic equation 9 Using (34) and estimation (32), we estimate in the following way: where . Hence and from (37) we get the estimation ( ) .Differentiating the both sides of (37) with respect to y , we have From (34) it follows that Obviously, for any natural number i .Knowing estimation (32) for Taking into account (38) and ( 41) in (40), we have  are the constants independent of ε .

Conclusion
Combining the obtained results, we arrive at the following statement.Theorem.Let ) , ( y x f be a function given in Π , have continuous derivatives with respect to x to the ( 1 + n )-th order inclusively, be infinitely differentiable with respect to y and satisfy equation (7).Then for the generalized solution of problem (1), (2) it holds asymptotic representation (51), where the functions i W are determined by the first iterative process, j V is a boundary layer type function near the boundary , 1 = x z is a remainder term and estimation (55) is valid for it.
Let's construct a boundary layer type function near the boundary 1 = x .The first iterative process is conducted on the base of decomposition (1) of the operator ε L .For conducting the second iterative process by means of which we'll construct a boundary layer function near the boundary 1 = x, it is necessary to write a new decomposition of the operator ε L near this boundary.We make change of variables: powers of ε in the coordinates ( y , τ ) has the form: get a new expansion of the function W in powers of ε in the coordinates ( ) τ , and the remaining functions k ω are determined from the formula obtained from equation (22) by differentiating with respect to y : ε .From (2), (24), (49), (50) and (51) it follows that z satisfies the boundary conditions: parts allowing for boundary conditions (54), after some transformations we get the estimation