Boundary layer solutions to singularly perturbed quasilinear systems (Q2090354)

This paper considers weak boundary layer solutions to the singularly perturbed ODE systems of the type \[\varepsilon^2(A(x,u(x),\varepsilon)u'(x))'=f(x,u(x),\varepsilon).\] The new features are that the authors do not consider one scalar equation, but systems, where the systems are allowed to be quasilinear, and that the systems are spatially non-smooth. Although the results about existence, asymptotic behavior, local uniqueness and stability of boundary layer solutions are similar to those known for semilinear, scalar and smooth problems, there are at least three essential differences: 1) the asymptotic convergence rates valid for smooth problems are not true anymore, in general, in the non-smooth case; 2) a specific local uniqueness condition from the scalar case is not sufficient anymore in the vectorial case; 3) the monotonicity condition, which is sufficient for stability of boundary layers in the scalar case, must be adjusted to the vectorial case