Ancient Solutions to the Biharmonic Heat Equation
Alexander D. McWeeney
TL;DR
This paper characterizes the space of ancient solutions to the biharmonic heat equation on certain manifolds, linking it to polynomially bounded biharmonic functions, thus generalizing previous results for caloric functions.
Contribution
It extends the understanding of ancient solutions from caloric to biharmonic heat equations on manifolds with polynomial volume growth.
Findings
Bounded the space of ancient solutions by dimensions of polynomially bounded biharmonic functions
Generalized previous work on caloric functions to biharmonic heat equations
Applicable to complete manifolds with polynomial volume growth
Abstract
We show that the space of polynomially bounded ancient solutions to the biharmonic heat equation on a complete manifold with polynomial volume growth is bounded by the dimensions of spaces of polynomially bounded biharmonic functions. This generalizes the work of Colding and Minicozzi in [6] for ancient caloric functions.
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Taxonomy
TopicsGeometry and complex manifolds · Geometric Analysis and Curvature Flows · Nonlinear Partial Differential Equations