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The maximum principle for systems of parabolic equations subject to an avoidance set. (English) Zbl 1049.35101
Hamilton’s maximum principle for systems states that given a reaction-diffusion equation (semi-linear heat-type equation) for sections of a vector bundle over a manifold, if the solution is initially in a subset invariant under parallel translation and convex in the fibers and if the ODE associated to the PDE preserves the subset, then the solution remains in the subset for positive time. The authors generalize this result to the case where the subsets are time-dependent and where there is an avoidance set from which the solution is disjoint. In applications the existence of an avoidance set can sometimes be used to prove the preservation of a subset of the vector bundle by the PDE.

35K57 Reaction-diffusion equations
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
35B50 Maximum principles in context of PDEs
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