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Uniqueness of nonnegative solutions of the Cauchy problem for parabolic equations on manifolds or domains. (English) Zbl 1024.35010

The uniqueness of nonnegative solutions of the Cauchy problem for parabolic equations on non-compact Riemannian manifolds or domains in \(\mathbb{R}^n\) is studied. The authors introduce two notions: the parabolic Harnack principle with scale function \(\rho\) concerning inhomogeneity at infinity of manifolds and the second order terms of equations; and the relative boundedness with scale function \(\rho\) concerning growth order at infinity of the lower terms of equations. In terms of this scale function, a general and sharp sufficient condition for the uniqueness of nonnegative solutions is given. A Täcklind type uniqueness theorem for solutions with growth conditions (which plays a crucial role in establishing a Widder-type uniqueness theorem for nonnegative solutions) is given, too. This result is new even for parabolic equations on \(\mathbb{R}^n\) in regard to growth rates at infinity of their lower order terms.

MSC:

35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
58J35 Heat and other parabolic equation methods for PDEs on manifolds
35K10 Second-order parabolic equations
31B35 Connections of harmonic functions with differential equations in higher dimensions
31C12 Potential theory on Riemannian manifolds and other spaces
53C20 Global Riemannian geometry, including pinching
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