Uniqueness/nonuniqueness for nonnegative solutions of a class of second-order parabolic equations
In this paper, we study the relationship between the long time behavior of a solution of the nonlinear heat equation on (where ) and the asymptotic behavior as of its initial value . In particular, we show that if the sequence of dilations converges weakly to as , then the rescaled solution converges uniformly on to along the subsequence , where is an appropriate flow. Moreover, we show there exists an initial value such that the set of all possible attainable in this...
We prove a generalization of the Douady-Oesterlé theorem on the upper bound of the Hausdorff dimension of an invariant set of a smooth map on an infinite dimensional manifold. It is assumed that the linearization of this map is a noncompact linear operator. A similar estimate is given for the Hausdorff dimension of an invariant set of a dynamical system generated by a differential equation on a Hilbert manifold.