A Singularly Perturbed Nonlinear Boundary Value Problem Arising In General Riemann Problem

In this paper we study a boundary value problem for a second order nonlinear ordinary differential equation involving a small parameter, 0, of the formon the whole real axis R with the boundary conditionsThis problem arises in the search of similarity solutions of the general Riemann problemWhere H(s) = 0 for s < 0, H(s) = 1 for s >0, and H(0) = [0,1], is the multiple valued Heaviside function. Riemann problem takes important station in certain scalar conservation law of physics.Here are our basic hypotheses:(H1) A, B,A< B,are given constants.(H2) Both f(t) and k(t) are real-valued continuous functions defined on [A,B], and f(A) = f(B) = 0,k(t) 0 a.e. on [A, B], (t) is an increasingcontinuous function defined on R, and (0) = 0, (+ ) = +, denoted its inverse function by -1(?.(H3) G(t) = I g(s)ds and 3>(t) = I ip(s)ds are both absolutely cori-tinuous functions defined on [A,B], and G(t) is strictly increasing on [A B]. (H4) There is a positive number M such that(H5) Suppose that w0 =: 0 on [A, B],G(t) - G(B) < f(t) < 0 or 0 < f(t) < G(t) on [A, B}. Under the above basic hypotheses we (formally) convert the boundary value problem (l?, (2) into the two-point value problem as follow:As the two endpoints t = A and t = B are singular for the problem, we need to consider the two-point boundary value problem (3e), (4/J. After discussing the properties of the solution sufficiently, we get the following results:THEOREM 1. Under the hypotheses (H1) - (H5), the two-point boundary value problem (3e), (40) with 0 has a unique solution w?t); the functionis locally absolutely continuous and strictly increasing in (A, B) when 0. Moreover, as e tends to zero, w (t) converges to 0(t) uniformly on [A, B] and y (t) converges to yo(t) uniformly on [A + 6,B - ] for any 2 (0, B - A). Then we construct the solution v (s) to the boundary value problem (l ), (2), utilizing the solution (t) of the boundary value problem (3 ), (40).In this section we mainly use the conclusion of the converse funtion and provedthat v?s) is a solution of (1? as k(t) > 0. Further more we discussed thestructure of the ve(s) when k(t) has at least one interval of degeneracy in[A,B].Summarized above we get the main results of the paper as following THEOREM 2. Assume that the hypotheses (H1) - (H5) are valid.Then the boundary value problem (le), (2), 0, has unique solution v?s).Moreover, the solution v(s) converges to the solution V0(S) pointwise on Ras 0.THEOREM 3. Let v0(s) be the solution to the problem (10), (2). ThenV0(S) can be represented bywhere v'0(s), a derivative of v0(s), is nonnegative and integrable on R, [s = Sj,j = 1,2, } the set of all jump points of V0(S), and {[aj, bj];j = 1,2, } the collection of all intervals of degeneracy of k(t) in [A, B]. Moreover, ineach connected component of the open set -R\U{s = Sj}, v0(s) is absolutelycontinuous and satisfies equation (10), while at each jump point s = Sj,j =1,2, , V0(S) must satisfy the following jump conditions (17) and (18).At the end of this paper, we deal with the follow singulary perturbed boundary value problemby using the same way and get the similary results.