Gradient estimates for a nonlinear parabolic equation under Ricci flow
نویسنده
چکیده
Let (M,g(t)), 0 ≤ t ≤ T , be a n-dimensional complete noncompact manifold, n ≥ 2, with bounded curvatures and metric g(t) evolving by the Ricci flow ∂gij ∂t = −2Rij . We will extend the result of L. Ma and Y. Yang and prove a local gradient estimate for positive solutions of the nonlinear parabolic equation ∂u ∂t = ∆u − au log u − qu where a ∈ R is a constant and q is a smooth function on M × [0, T ].
منابع مشابه
Gradient Estimates for a Nonlinear Parabolic Equation on Riemannian Manifolds
Let (M, g) be a complete noncompact Riemannian manifold. In this paper, we derive a local gradient estimate for positive solutions to a simple nonlinear parabolic equation ∂u ∂t = ∆u+ au log u+ bu on M × [0,+∞), where a, b are two real constants. This equation is closely related to the gradient Ricci soliton. We extend the result of L. Ma (Journal of Functional Analysis 241 (2006) 374-382).
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