An identification problem governed by nonlinear fractional mobile-immobile equation, Part II: Stability and Regularity
DOI:
https://doi.org/10.56764/hpu2.jos.2025.4.03.42-50Abstract
This paper continues our recent work in [1], in which we investigated the existence and uniqueness of solutions for the inverse problem (IP): Seek the unknown term \(z\), along with the state \(u\) obeying the system
\[\nu_1{\partial_t}u+\nu_2\partial_t^\alpha u - \Delta u = z(x)h(t) + f(t,u), \text{ in }\Omega ,t \in (0,T], \,\tag{1}\]
\[u = 0\text{ on }\partial \Omega ,t \ge 0,\,\tag{2}\]
\[u(\cdot ,0)\text{ = }\xi \text{ in }\Omega ,\,\tag{3}\]
and the terminal measurement
\[u(x,T)=\varphi (u)(x), x \in \Omega . \,\tag{4}\]
In this setting, \(\Omega\subset\mathbb{R}^d\) with \(d \ge 1\) denotes a bounded domain whose boundary \(\partial\Omega\) is smooth. Motivated by considerations arising from numerical analysis, the primary aim of this work is to establish a set of sufficient conditions on the functions \(h\), \(f\), and \(\varphi\) that guarantee both the continuous dependence of solutions on the data and the regularity in time of the solution pair \((u,z)\) for the inverse problem (IP).
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Copyright (c) 2025 Van-Tuan Tran, Thi-Thu Tran , Van-Dac Nguyen
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