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This talk is concerned with several eigenvalue problems from a linear stability analysis for the steady state morphogen gradients in Drosophila wing imaginal discs. These problems share several common difficulties including the followings: 1) The steady state solution which occurs in the coefficients of the relevant differential equations of the stability analysis is only know qualitatively and numerically. 2) Though the governing differential equations are linear, the eigenvalue parameter appears nonlinearly in the differential equations as well as in the boundary conditions. 3) The eigenvalues are determined not only as solutions of a homogeneous boundary value problem in differential equations with homogeneous boundary conditions of the Dirichlet type, they also come from a condition arising from a boundary condition of the original problem complementary to the Dirichlet condition.

This talk reports on two principal results for these nonlinear eigenvalue problems. Regarding the stability of the steady state morphogen gradients, we prove that the eigenvalues must all be positive and hence the steady state morphogen gradients are asymptotically stable. In addition, we will report on a novel result pertain to the determination of the smallest (positive) eigenvalue that determines the decay time of transients and the time needed to reach steady state. Here we prove that the smallest eigenvalue does not come from the nonlinear Dirichlet eigenvalue problem and we need only to find the smallest root of a relevant polynomial. Keeping in mind that even the steady state solution needed for the stability analysis is only known numerically, not having to solve the nonlinear Dirichlet eigenvalue problem is both an attractive theoretical outcome and a huge computational simplification.

The work reported was done jointly the speakers colleague, Professor Qing Nie, at UC Irvine and Professor Yuan Lou of the Department of Mathematics at the Ohio State University. The work of Professor Nie and the speaker was supported by NIH Grant R01- GM67247.