Original Title: “Approximation theorems for parabolic equations and movement of local hot spots”
Authors: Alberto Enciso (ICMATCSIC), María Ángeles García-Ferrero (Universidad de Heidelberg) y Daniel Peralta-Salas (ICMATCSIC)
Source: Duke Math. J. 168 (2019), 897939
Date of online publication: 2019
The theory of global approximation for (linear) elliptic PDE was developed during the period 19501965 by Browder, Lax and Malgrange, as a generalization of the classical Runge’s theorem in complex analysis. Roughly speaking, it states that given a solution to the PDE defined on a closed set K, there exists a global solution that approximates it, provided that K satisfies some mild topological assumptions. For parabolic PDE this theory had been developed only for the usual heat equation in Euclidean space and compact sets K. In this paper the global approximation theory for parabolic PDE is developed with the same generality as the elliptic case: parabolic operators with Holder nonconstant coefficients and closed sets Knot necessarily compact.
In the case of the heat equation in Euclidean space, it is shown that in fact the global solution corresponds to a smooth Cauchy datum with compact support. The proofs are technically quite involved and make use of the fundamental solution of the parabolic PDE and several tools from functional analysis, harmonic analysis and parabolic Schauder estimates. One of the most striking applications of the theory developed in this work is the analysis of the hot spots of the solutions to the heat equation, which is a topic that has attracted an enormous attention in the last years. It is shown that there exist global solutions exhibiting a local hot spot that moves along a prescribed curve for all time, up to a small error that tends to zero as time grows. As a corollary, it is proved that there exist solutions with a hot spot that fills densely the whole space. Other remarkable application is in the context of isothermic surfaces, i.e. surfaces where the solution is constant (that usually depends on time). It is shown that there exist solutions with isothermic surfaces that change their topology in any prescribed a priori way.