SHOCK LAYER BEHAVIOR FOR VECTOR BOUNDARY-VALUE-PROBLEMS AND KINKS IN THE LOWER AND UPPER SOLUTIONS

[摘要] We provide sufficient conditions for the existence and approximations of shock layer solutions of the weakly coupled boundary value problem epsilony(i)''= f(i)(t, y)y(i)' + g(i)(t, y) i = 1, ..., n, y(i)(a) = A(i), y(i)(b) = B(i), where y, f and g are n-dimensional real-valued functions and y(i), f(i), and g(i) are the respective vector components. epsilon is a small positive parameter. Our approach is based upon the construction of lower and upper ''solutions'' which have boundary layer behavior at specific points T(i) (i=1, ..., n) in the interval a< t < b. By adjoining these lower and upper ''solutions'' in an appropriate fashion, we are able to obtain lower and upper solutions defined on the entire interval a less-than-or-equal-to t less-than-or-equal-to b. In particular, an analysis is presented that insures the lower and upper solutions kink in the correct directions in the regions near the shock layers. The details of this analysis have often been left out of studies of interior layer behavior when using differential inequalities and have contributed to published works where the necessary kinks are in the wrong direction in the proofs. It also appears that a sign restriction is needed on the derivative of the shock layer function to ensure that the lower and upper solutions do not intersect. We believe this restriction to be a consequence of using differential inequalities, as positive results have been obtained for the scalar case using at least two other methods, namely continuity arguments and contraction mappings. Unfortunately, the vector problem seems to be either intractable or extremely tedious when using those methods. (C) 1994 Academic Press, Inc.

[发布日期] 1994-07-01