AI 中文总结
本文研究任意维有界光滑区域上带齐次Neumann边界条件的离散扩散交换驱动增长方程，对特定可分离交换核建立了全局时间解的存在性。
AI 中文摘要
我们研究任意维有界光滑区域上的离散扩散交换驱动增长（EDG）方程，该方程服从齐次Neumann边界条件。该系统属于具有二次型非线性源项的无穷半线性偏微分方程组类。对于形如\boldsymbol{K_{i,j}=b_i a_j}的可分离交换核，当供体率至多线性增长、受体率次线性增长时，我们建立了解的全局时间存在性。分析基于从熵-熵耗散恒等式得到的一致Fisher信息估计，该估计为具有有限多个物种的截断系统提供了重整化解。随后，紧性论证使得我们可以在交换算子中取极限，从而得到完整系统的全局时间解的存在性。
英文摘要
We study the discrete diffusive exchange-driven growth (EDG) equations on a bounded smooth domain of arbitrary dimension subject to homogeneous Neumann boundary conditions. The system belongs to the class of infinite systems of semilinear partial differential equations with nonlinear source terms of quadratic type. Global-in-time existence of solutions is established for separable exchange kernels of the form \(K_{i,j}=b_i a_j\), where the donor rates exhibit at most linear growth while the receiver rates are sublinear. The analysis is based on a uniform Fisher information estimate obtained from an entropy-entropy dissipation identity. This estimate yields renormalized solutions to a truncated system with finitely many species. A compactness argument then enables passage to the limit in the exchange operator, leading to the existence of global-in-time solutions for the full system.
Comments33 pages
