A system of drift-diffusion equations for the electron, hole, and oxygen vacancy densities in a semiconductor, coupled to the Poisson equation for the electric potential, is analyzed in a bounded domain with mixed Dirichlet–Neumann boundary conditions. This system describes the dynamics of charge carriers in a memristor device. Memristors can be seen as nonlinear resistors with memory, mimicking the conductance response of biological synapses. In the fast-relaxation limit, the system reduces to a drift-diffusion system for the oxygen vacancy density and electric potential, which is often used in neuromorphic applications. The following results are proved: the global existence of weak solutions to the full system in any space dimension; the uniform-in-time boundedness of the solutions to the full system and the fast-relaxation limit in two space dimensions; the global existence and weak–strong uniqueness analysis of the reduced system. Numerical experiments in one space dimension illustrate the behavior of the solutions and reproduce hysteresis effects in the current–voltage characteristics.

在具有混合狄利克雷-诺依曼边界条件的有界域中分析了半导体中电子、空穴和氧空位密度的漂移扩散方程组，与电位的泊松方程耦合。该系统描述了忆阻器器件中电荷载流子的动态。忆阻器可以看作是具有记忆功能的非线性电阻器，模仿生物突触的电导响应。在快速弛豫极限下，系统简化为氧空位密度和电势的漂移扩散系统，这通常用于神经形态应用。证明了以下结果：整个系统在任何空间维度上都存在弱解；整个系统解的时间一致有界性和二维空间维度的快速松弛极限；简化系统的全局存在性和弱强唯一性分析。一维空间中的数值实验说明了解决方案的行为并再现了电流-电压特性中的磁滞效应。