摘要
This paper studies the boundary value problem involving a small parameter(（k（V（t））+ε）[V’（s）[<sup>n-1V’（s）</sup>’+(sg（V（s））+f（V（s））V’（s）=0 for s∈R,V（-∞）=A,V（+∞）=B;A【B,which originates from the Riemann problem for a generalized diffusion equntiong（U）D<sub>t</sub>U=p’（t）p<sup>N</sup>（t）D<sub>x</sub>(（k（U）+ε）|D<sub>x</sub>U|<sup>N-1D<sub>x</sub>U</sup>+p’（t）f（U）D<sub>x</sub>U for x∈R,t】0,U（x,0）=A for x【0,U（x,0）=B for x】0,under the hypotheses H<sub>1</sub>—H<sub>4</sub>.The author’s aim is not only to determine explicitly thediscontinuous solution U<sub>0</sub>（x,t）=V<sub>0</sub>（s）,s=x/p（t）,to the reduced problem,and the formand the number of its curves of discontinuity,but also to present,in an extremely naturalway,the jump conditions which it must satisfy on each of its curves of discontinuity.Itis proved that the problem has a unique solution U<sub>ε</sub>（x,t）=V<sub>ε</sub>（s）,s=x/p（t）,ε≥0,V<sub>ε</sub>（s）pointwise converges to V<sub>0</sub>（s）as ε 0,V<sub>0</sub>（s）has at least one jump point if and only if k（y）possesses at least one interval of degeneracy in[A,B],and there exists a one-to-onecorrespondence between the collection of all intervals of degeneracytof k（y）in[A,B]andthe set of all jump points of V<sub>0</sub>（s）.
This paper studies the boundary value problem involving a small parameter ((k(V(t))+epsilon)\V'(s)\N-1V'(s))'+(sg(V(s))+f(V(s)))V'(s) = 0 for s subset-of R, V(- infinity) = A, V(+ infinity) = B; A < B, which originates from the Riemann problem for a generalized diffusion equntion g(U)D(t)U = p'(t)p(N)(t)D(x)((k(U)+ epsilon)\D(x)U\N-1D(x)U)+p'(t)f(U)D(x)U for x subset-of R, t > 0, U(x,0) = A for x < 0, U (x,0) = B for x > 0, under the hypotheses H1-H4. The author's aim is not only to determine explicitly the discontinuous solution U0(x, t) = V0(s), s = x/p(t), to the reduced problem, and the form and the number of its curves of discontinuity, but also to present, in an extremely natural way, the jump conditions which it must satisfy on each of its curves of discontinuity. It is proved that the problem has a unique solution U-epsilon(x,t) = V-epsilon(s), s = x/p(t), 8 greater-than-or-equal-to 0, V-epsilon(s) pointwise converges to V0(s) as epsilon arrow-pointing-down 0, V0(s) has at least one jump point if and only if k(y) possesses at least one interval of degeneracy in [A, B], and there exists a one-to-one correspondence between the collection of all intervals of degeneracy of k(y) in [A, B] and the set of all jump points of V0(s).
