确定时间源因子和具有未知初值的分数阶的唯一性

中文摘要 AI 辅助

我们考虑时间分数阶扩散方程的反问题，该方程具有二阶椭圆算子，并在$\\(\mathbb R^N\\)$的有界域中满足齐次狄利克雷边界条件。初值是未知的，且不假设为零。主要结果涉及从一次加权空间观测中同时确定分数阶和时间相关源因子。对于$(0,1)$中的无理数阶和满足指数可积条件的有界源因子，我们证明对所有正时间给出的非零观测唯一地确定这两个未知量。我们还描述了由该观测确定的初值信息。一个反例表明，在经典阶$\\(\alpha=1\\)$时，即使观测非零，源因子的唯一性也可能失效。在第二部分，我们建立了当比较方程中的椭圆算子、初值和源项可能不同时，分数阶的两个唯一性结果。第一个结果通过拉普拉斯变换证明，而第二个结果使用短时观测和Mittag--Leffler函数的渐近行为。

英文摘要

We consider inverse problems for a time-fractional diffusion equation with a second-order elliptic operator and the homogeneous Dirichlet boundary condition in a bounded domain of $\mathbb R^N$. The initial value is unknown and is not assumed to be zero. The main result concerns the simultaneous determination of the fractional order and the time-dependent source factor from one weighted spatial observation. For irrational orders in $(0,1)$ and bounded source factors satisfying an exponential integrability condition, we prove that a nonzero observation given for all positive times uniquely determines both unknowns. We also describe the information about the initial value determined by this observation. A counterexample shows that uniqueness of the source factor may fail at the classical order $α=1$, even when the observation is nonzero. In the second part, we establish two uniqueness results for the fractional order when the elliptic operators, initial values, and source terms in the compared equations may be different. The first result is proved by the Laplace transform, whereas the second uses short-time observations and the asymptotic behavior of the Mittag--Leffler functions.

发表机构

• V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences（乌兹别克斯坦科学院 V.I.罗曼诺夫斯基数学研究所）
• School of Engineering, Central Asian University（中亚大学工程学院）
• Graduate School of Mathematical Sciences, The University of Tokyo（东京大学数理科学研究科）
• Department of Mathematics, Faculty of Arts and Sciences, Zonguldak Bülent Ecevit University（宗古尔达克布伦特·埃切维特大学文理学院数学系）

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22 pages