非紧Riemann流形上一类Kazdan-Warner型方程光滑解的存在唯一性

Existence and Uniqueness of Smooth Solution to a Class of Kazdan-Warner Typed Equations on Non-Compact Riemannian Manifolds

在非紧Riemann流形上讨论了一类Kazdan-Warner型方程。首先,利用穷竭法以及标准的抛物理论得到了一类带初始条件和Neumann边界条件的热流方程长时间解的存在唯一性。然后得到了热流方程解的一致估计并在合适的条件下得到了所讨论方程光滑解的存在唯一性。

A class of Kazdan-Warner typed equations on non-compact Riemannian manifolds are discussed.Firstly,the existence and uniqueness of longtime solution to the corresponding heat equation with initial condition and Neumann boundary condition are obtained by the exhaustion method and the standard parabolic theory.And then,some uniform estimates of the solution to the heat equation are obtained and the existence and uniqueness of smooth solution to the equations discussed in this paper are gained under suitable conditions.