Singular perturbation of the fourth order elliptic equation when the limit equation is elliptic-parabolic

Abstract

In this paper we cosider the singular perturbation of the fourth order elliptic equation$- \varepsilon ^2 \Delta ^2 u + y^m \frac{{\partial ^2 u}}{{\partial y^2 }} + \frac{{\partial ^2 u}}{{\partial x^2 }} + a(x, y)\frac{{\partial u}}{{\partial y}} + b(x, y)\frac{{\partial u}}{{\partial x}} + c(x, y) = 0$ when the limit equation is elliptic-parabolic, where ε is a positive parameter, m is a positive real number, Δis Laplacian operator, a,b,c are sufficiently smooth. Under appropriate condition we derive the sufficient condition of solvability and prove the existence of solution and give a uniformly valid asymptotic solution of arbitrary order.