Nonlinear Eigenvalue Problems under Strong Localized Perturbations with Applications to Chemical Reactors

摘要

Nonlinear eigenvalue problems are considered for partial differential equations and boundary conditions of the forThe problems are perturbed by deleting a small subdomainDεfromDand imposing a boundary condition on the surface of the resulting hole, or else by changing the constantbin the boundary condition to a different constantε−1kon a small part of∂D. In both cases the perturbed solution is constructed forεsmall by the method of matched asymptotic expansions. Particular attention is paid to the calculation ofλc(ε), the critical value ofλat which the number of solutions changes by two, for example from two to none. This value ofλrepresents a simple fold point in the bifurcation diagram of ‖u‖ versusλ. The analysis is applied to chemical reactors withF(x,u)=eu, in which caseλcis the critical value of the Frank‐Kamenetskii parameter. The asymptotic results are compared with exact and numerical results for some special cases, and fair agreement is found. Some previous asymptotic calculations are fou