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Parabolic equations for curves on surfaces. I: Curves with \(p\)-integrable curvature. (English) Zbl 0789.58070
Summary: This is the first of a two-part paper in which we develop a theory of parabolic equations for curves on surfaces which can be applied to the so-called curve shortening of flow-by-mean-curvature problem, as well as to a number of models for phase transitions in two dimensions.
We introduce a class of equations for which the initial value problem is solvable for initial data with \(p\)-integrable curvature, and we also give estimate for the rate at which the \(p\)-norms of the curvature must blow up, if the curve becomes singular in finite-time.
A detailed discussion of the way in which solutions can become singular and a method for “continuing the solution through a singularity” will be the subject of the second part.

MSC:
58J35 Heat and other parabolic equation methods for PDEs on manifolds
35K15 Initial value problems for second-order parabolic equations
35K45 Initial value problems for second-order parabolic systems
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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