Proving the uniqueness of solutions to multi-species cross-diffusion systems is a difficult task in the general case, and there exist very few results in this direction. In this work, we study a particular system with zero-flux boundary conditions for which the existence of a weak solution has been proven in Ehrlacher and Bakhta (ESAIM Math Model Numer Anal, 2017). Under additional assumptions on the value of the cross-diffusion coefficients, we are able to show the existence and uniqueness of non-negative strong solutions. The proof of the existence relies on the use of an appropriate linearized problem and a fixed-point argument. In addition, a weak–strong stability result is obtained for this system in dimension one which also implies uniqueness of weak solutions

中文翻译：

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交叉扩散方程组中强解和弱强稳定性的唯一性

在一般情况下，证明多物种交叉扩散系统解决方案的唯一性是一项艰巨的任务，而在此方向上几乎没有结果。在这项工作中，我们研究了具有零通量边界条件的特定系统，该系统在Ehrlacher和Bakhta中已经证明了弱解的存在（ESAIM Math Model Numer Anal，2017）。在关于交叉扩散系数的值的其他假设下，我们能够证明非负强解的存在性和唯一性。存在的证明依赖于适当线性化问题和定点参数的使用。此外，该系统在第一维上获得了弱-强稳定性结果，这也暗示了弱解的唯一性