• Bulletin of the Korean Mathematical Society
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• v.35 no.2
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• pp.387-395
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• 1998

In this article we prove a Harnack inequality for the positive solutions of the heat equation with Neumann boundary condition for a compact Riemannian manifold with possibly non-convex boundary.

• Bulletin of the Korean Mathematical Society
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• v.50 no.5
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• pp.1587-1598
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• 2013

In this paper, we obtain some gradient estimates for positive solutions to the following nonlinear parabolic equation $$\frac{{\partial}u}{{\partial}t}={\triangle}u-b(x,t)u^{\sigma}$$ under general geometric flow on complete noncompact manifolds, where 0 < ${\sigma}$ < 1 is a real constant and $b(x,t)$ is a function which is $C^2$ in the $x$-variable and $C^1$ in the$t$-variable. As an application, we get an interesting Harnack inequality.

• Journal of the Korean Mathematical Society
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• v.33 no.2
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• pp.455-468
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• 1996

Certain analytic behavior of geometric objects defined on a Riemannian manifold depends on some very crude properties of the manifold. Some of those crude invariants are the volume growth rate, isoperimetric constants, and the likes. However, these crude invariants sometimes exercise surprising control over the analytic behavior.

• Bulletin of the Korean Mathematical Society
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• v.37 no.2
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• pp.303-316
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• 2000

The Cauchy problem for degenerate parabolic equations with absorption is studied. We prove the existence of a fundamental solution. Also a Harnack type inequality is established and the existence and uniqueness of initial trace for nonnegative solutions is shown.

• Journal of the Korean Mathematical Society
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• v.61 no.2
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• pp.377-393
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• 2024

In this paper, we study the positive solutions to a discrete harmonic function for a random walk satisfying finite range and ellipticity conditions, killed at the boundary of an unbounded cylinder in ℤ^d. We first prove the existence and uniqueness of positive solutions, and then establish that all the positive solutions are generated by two special solutions, which are exponential growth at one end and exponential decay at the other. Our method is based on maximum principle and a Harnack type inequality.

• East Asian mathematical journal
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• v.30 no.1
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• pp.79-91
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• 2014

We consider the parabolic evolution differential equation such as heat equation and porus-medium equation on a Riemannian manifold M whose Ricci curvature is bounded below by $-(n-1)k^2$ and bounded below by 0 on some amount of M. We derive some bounds of differential quantities for a positive solution and some inequalities which resemble Harnack inequalities.

• Journal of the Korean Mathematical Society
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• v.36 no.1
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• pp.73-95
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• 1999

We prove that if a given complete Riemannian manifold is roughly isometric to a complete Riemannian manifold satisfying the volume doubling condition, the Poincar inequality and the finite covering condition at infinity on each end, then every positive harmonic function on the manifold is asymptotically constant at infinity on each end. This result is a direct generalization of those of Yau and of Li and Tam.

• Journal of the Korean Mathematical Society
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• v.57 no.6
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• pp.1573-1590
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• 2020

In this paper, we classify all solutions bounded from below to uniformly elliptic equations of second order in the form of Lu(x) = a_ij(x)D_iju(x) + b_i(x)D_iu(x) + c(x)u(x) = f(x) or Lu(x) = D_i(a_ij(x)D_ju(x)) + b_i(x)D_iu(x) + c(x)u(x) = f(x) in unbounded cylinders. After establishing that the Aleksandrov maximum principle and boundary Harnack inequality hold for bounded solutions, we show that all solutions bounded from below are linear comb_inations of solutions, which are sums of two special solutions that exponential growth at one end and exponential decay at the another end, and a bounded solution that corresponds to the inhomogeneous term f of the equation.

• Kyungpook Mathematical Journal
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• v.61 no.3
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• pp.631-644
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• 2021

In the present paper, we obtain gradient estimates for positive solutions to the following nonlinear parabolic equation under general geometric flow on complete noncompact manifolds $$\frac{{\partial}u}{{\partial}t}={\Delta}u+a(x,t)u^p+b(x,t)u^q$$ where, 0 < p, q < 1 are real constants and a(x, t) and b(x, t) are functions which are C² in the x-variable and C¹ in the t-variable. We shall get an interesting Harnack inequality as an application.

• Bulletin of the Korean Mathematical Society
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• v.58 no.4
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• pp.909-914
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• 2021

In this short note we prove new differential Harnack inequalities interpolating those for the static surface and for the Ricci flow. In particular, for 0 ≤ 𝜀 ≤ 1, α ≥ 0, 𝛽 ≥ 0, 𝛾 ≤ 1 and u being a positive solution to $${\frac{{\partial}u}{{\partial}t}}={\Delta}u-{\alpha}u\;{\log}\;u+{\varepsilon}Ru+{\beta}u^{\gamma}$$ on closed surfaces under the flow ${\frac{\partial}{{\partial}t}}g_{ij}=-{\varepsilon}Rg_{ij}$ with R > 0, we prove that $${\frac{\partial}{{\partial}t}}{\log}\;u-{\mid}{\nabla}\;{\log}\;u{\mid}^2+{\alpha}\;{\log}\;u-{\beta}u^{{\gamma}-1}+\frac{1}{t}={\Delta}\;{\log}\;u+{\varepsilon}R+{\frac{1}{t}{}\geq}0$$.