Institute of Math. and Mechanics Russian Academy of Sciences / Ural Branch
S.Kovalevskaya 16, GSP-384 Ekaterinburg 620219 RUSSIA

e-mails: pmf@imm.uran.ru
            Marina@vpro.convex.ru
http://vpro.convex.ru/Marina

Heat equation on Riemann manifolds: morphisms and reduction to smaller dimension

Abstract:
There is considered the category of nonlinear heat equations posed on Riemann manifolds. In this category, for given heat equation we could find morphisms from it to other equations with the same or smaller number of independent variables. It allows to receive some classes of solutions of original equation from the class of all solutions of such reduced equation. This concept is proposed in general case for the arbitrary PDE systems, and its concrete investigation is developing for the heat equation case.
There are derived the necessary and sufficient conditions for morphisms of heat equation to the heat equation on the other manifold. These conditions are formulated in the differential geometry language. Classification of morphisms (with selection from every equivalence class of the simplest representative) is carried out.
The comparison with invariant solutions classes, obtained by the Lie group methods, is carrying out. It is proved that discovering solution classes are richer than invariant solution classes, even if we find any (including discontinuous) symmetry groups of original equation.