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Sharp gradient estimates for a heat equation in Riemannian manifolds. (English) Zbl 1428.58021

Authors’ abstract: We prove sharp gradient estimates for a positive solution to the heat equation \(u_t=\Delta u+au\log u\) in complete noncompact Riemannian manifolds. As its application, we show that if \(u\) is a positive solution of the equation \(u_t=\Delta u\) and \(\log u\) is of sublinear growth in both spatial and time directions, then \(u\) must be constant. This gradient estimate is sharp since it is well known that \(u(x,t)=e^{x+t}\) satisfying \(u_t=\Delta u\).

MSC:

58J35 Heat and other parabolic equation methods for PDEs on manifolds
35K05 Heat equation
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
53C20 Global Riemannian geometry, including pinching
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