This article investigates the local well-posedness of Turing-type reaction–diffusion equations with Robin boundary conditions in the Sobolev space. Utilizing the Hadamard norm, we derive estimates for Fokas unified transform solutions for linear initial-boundary value problems subject to external forces. Subsequently, we demonstrate that the iteration map, defined by the unified transform solutions and incorporating nonlinearity instead of external forces, acts as a contraction map within an appropriate solution space. Our conclusive result is established through the application of the contraction mapping theorem.

中文翻译：

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反应扩散方程耦合系统的 Robin 问题

本文研究了 Sobolev 空间中具有 Robin 边界条件的图灵型反应扩散方程的局部适定性。利用 Hadamard 范数，我们得出了受外力影响的线性初始边值问题的 Fokas 统一变换解的估计。随后，我们证明了由统一变换解定义并结合非线性而不是外力的迭代图在适当的解空间内充当收缩图。我们的结论性结果是通过应用收缩映射定理得出的。