Affiliations: Sobolev Institute of Mathematics, 4 Acad. Koptyug prosp., 630090 Novosibirsk, Russia E‐mail: adr@math.nsc.ru | Institute of Automation and Electrometry, 1 Acad. Koptyug prosp., 630090 Novosibirsk, Russia E‐mail: mmlavr@nsu.ru | Dipartimento di Matematica, Università di “Roma Tre”, 1 Largo San Leonardo Murialdo, 00146 Rome, Italy E‐mail: spigler@mat.uniroma3.it
Note: [] Corresponding author. Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università di Padova, Via Belzoni, 7, 35131 Padova, Italy.
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.
