随机两物种趋化竞争：全局适定性与连续依赖性
Stochastic Two-Species Chemotaxis Competition: Global Well Posedness and Continuous Dependence
AI总结：
该研究针对有界光滑区域上的随机两物种趋化竞争Keller–Segel模型，证明自阻尼指数大于2时系统存在唯一全局非负温和解，且解对初值依概率连续依赖，完善了带噪声的多物种趋化系统适定性理论。
AI中文摘要：
本文研究一类随机两物种竞争Keller–Segel模型。当两个物种对同一化学信号作出响应时，趋化运动、种间竞争与环境波动可能同时发挥作用。我们在有界光滑区域$\u274c\u2282\ub211^n$（$n\in\{1,2\}$）上考虑一个随机两物种趋化-竞争系统。两个物种均促进信号的产生并沿信号梯度运动；它们的种群方程还包含非线性竞争项与乘性噪声。若自阻尼指数大于2，我们证明该系统存在唯一的全局适应非负温和解。我们还建立了有限时间区间上系统对容许初值依概率的连续依赖性。结果证明了在所述假设下的全局适定性，同时表明初始种群的依概率微小变化会导致解的微小变化。由于存在两个耦合的趋化通量与随机项，$L^p$估计中包含无固定符号的项，这些项通过信号方程的抛物估计、超二次自阻尼以及随机卷积估计得到控制。
英文摘要：
In this paper, we investigate a stochastic two species competition Keller--Segel model. Chemotactic movement, interspecific competition, and environmental fluctuations may act simultaneously when two species respond to the same chemical signal. We consider a stochastic two-species chemotaxis--competition system on a bounded smooth domain $\mathcal O\subset\mathbb R^n$, $n\in\{1,2\}$. Both species contribute to the production of the signal and move along its gradient; their population equations also include nonlinear competition and multiplicative noise. If the self-damping exponents exceed two, we prove that the system has a unique global adapted nonnegative mild solution. We also establish continuous dependence in probability on admissible initial data over finite time intervals. The results prove the global well-posedness under the stated assumptions and we still show that small changes in the initial populations, in probability, to small changes in the solution. Due to the two coupled chemotactic fluxes and the stochastic terms, the $L^p$ estimates contain terms with no fixed sign. They are controlled by parabolic estimates for the signal equation, superquadratic self-damping, and stochastic convolution estimates.
