ON ASYMPTOTIC “EIGENFUNCTIONS” OF THE CAUCHY PROBLEM FOR A NONLINEAR PARABOLIC EQUATION

The asymptotic () behavior of solutions of the Cauchy problem is studied for the semilinear parabolic equation where and as . The existence is established of an infinite collection (a continuum) of distinct self-similar solutions of the form , , where the function satisfies an ordinary differential equation. Conditions for the asymptotic stability of these solutions are established. It is shown that for there exist solutions of the problem whose behavior as is described by approximate self-similar solutions (ap.s.-s.s.'s) which in the case coincide with a family of self-similar solutions of the heat equation , while for and the ap.s.-s.s. has the form where .Figures: 2. Bibliography: 78 titles.

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