Singular perturbations for certain partial differential equations without boundary‐layers

Abstract

Linear parabolic partial differential equations with a small parameter multiplying some of the higher space derivatives are considered, in the limiting case when such parameter vanishes. A number of different boundary‐value problems for singularly perturbed equations are examined. Such problems are unified by the rather unexpected property that no boundary‐layers are required, despite of the presence of the small vanishing parameter. The high points are the following. First, solvability theorems for some classes of ultraparabolic problems have been established. Second, the boundary conditions to be imposed to obtain well‐posed problems do not depend on the sign of the coefficient multiplying the time‐like derivative. Third, all coefficients are allowed to depend on all space and time variables. These results, in part, have been established by imposing suitably generalized compatibility conditions on coefficients and data.