A Source Identification Problem for the Bi-Parabolic Equation Containing a Poly-harmonic Operator

In this paper, we address the source identification problem for the bi-parabolic equation involving a operator. Specifically, we investigate the equation $({\mathrm{\partial }}_t+\mathfrak{A}{)}^{2}u(t)=\psi (t)f$, where $\mathfrak{A}$ denotes a poly-harmonic operator. Given the perturbed data of $\psi$ and $u(T)$ (where $T>0$), our objective is to determine $f$. Although several scientific publications have explored regularization techniques for bi-parabolic problems, the existing literature remains limited. By relaxing certain conditions on the function $\psi$ and employing a truncation regularization method while considering the problem on an unbounded domain, we believe our results provide valuable insights.