This paper focus on the problem of reconstructing the time-dependent potential in a class of generalized (including three special cases: the classical/multi-term/distributed order) time-fractional super-diffusion equations with nonlinear sources from a nonlocal integral observation. For such nonlinear equation, we investigate it for both the direct and inverse time-dependent potential problems. For the direct problem, given the time-dependent potential function , we obtain the well-posedness of the corresponding initial–boundary value problem. For the inverse potential problem, by utilizing additional nonlocal measurement data and using the Arzelà–Ascoli theorem and Grönwall’s inequality, we prove the existence and uniqueness of the solution for such nonlinear problem. Meanwhile, we also demonstrate the ill-posedness of the inverse problem. Moreover, to validate the theoretical results, we numerically reconstruct the potential term from Bayesian perspective. Several numerical examples are presented to show the efficiency of the proposed method.

中文翻译：

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根据非局部积分观测值反演具有非线性源的广义时间分数超扩散方程的时间相关潜在问题


本文重点研究一类广义（包括三种特殊情况：经典/多项/分布阶）时间分数超扩散方程中从非局部积分观测到的非线性源重建随时间势的问题。对于这种非线性方程，我们研究了它的正向和逆向时间相关的潜在问题。对于直接问题，给定与时间相关的势函数 ，我们获得相应的初始边值问题的适定性。对于逆势问题，利用额外的非局部测量数据，并利用Arzelà-Ascoli定理和Grönwall不等式，证明了该非线性问题解的存在性和唯一性。同时，我们还证明了反问题的不适定性。此外，为了验证理论结果，我们从贝叶斯角度对势项进行了数值重建。几个数值例子显示了所提出方法的效率。