We prove existence and uniqueness of solutions to the initial-boundary value problem for the Lifshitz–Slyozov equation (a nonlinear transport equation on the half-line), focusing on the case of kinetic rates with unbounded derivative at the origin. Our theory covers in particular those cases with rates behaving as power laws at the origin, for which an inflow behavior is expected and a boundary condition describing nucleation phenomena needs to be imposed. The method we introduce here to prove existence is based on a formulation in terms of characteristics, with a careful analysis on the behavior near the singular boundary. As a byproduct we provide a general theory for linear continuity equations on a half-line with transport fields that degenerate at the boundary. We also address both the maximality and the uniqueness of inflow solutions to the Lifshitz–Slyozov model, exploiting monotonicity properties of the associated transport equation.

我们证明了 Lifshitz-Slyozov 方程（半线上的非线性输运方程）初边值问题解的存在性和唯一性，重点关注原点导数无界的动力学速率的情况。我们的理论特别涵盖了那些速率在原点表现为幂律的情况，对于这些情况，预期流入行为并且需要施加描述成核现象的边界条件。我们这里介绍的证明存在的方法是基于特征的公式化，并对奇异边界附近的行为进行仔细分析。作为副产品，我们提供了半线上线性连续方程的一般理论，其中输运场在边界处退化。