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 ${({\partial }_t+A)}^{2}u\left(t\right)=f\cdot \psi \left(t\right)$, where $A$ denotes a poly-harmonic operator. Given the perturbed data of ψ and $u\left(T\right)$ (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 ψ and employing a truncation regularization method while considering the problem on an unbounded domain, we believe our results provide valuable insights.