Matrix Harnack estimates for a parabolic equation on compact K\"ahler manifolds
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We prove matrix and classical Harnack estimates for a positive solution to the parabolic equation \[ \partial_tu(x,t)=\Delta u(x,t)+A(u(x,t)) \] on a compact K\"ahler manifold. By tracing and integrating the matrix Harnack inequality, we obtain the classical Harnack estimate. We also give matrix and classical Harnack inequalities of two classes of specific parabolic equations, which extend existing results to more general cases.
Abstract
We prove matrix and classical Harnack estimates for a positive solution to the parabolic equation \[ \partial_tu(x,t)=\Delta u(x,t)+A(u(x,t)) \] on a compact K\"ahler manifold. By tracing and integrating the matrix Harnack inequality, we obtain the classical Harnack estimate. We also give matrix and classical Harnack inequalities of two classes of specific parabolic equations, which extend existing results to more general cases.