Backward Homogeneous Heat Equation with $\mathrm{\Psi }$-Caputo Time Derivative and Spectral Fractional Laplacian

In this work, we study a fractional diffusion model in time and space, formulated in an open and bounded domain, involving a Caputo fractional derivative of order $\mu \in (0,1)$ with respect to a time scale function $\mathrm{\Psi }$, together with a spectral fractional Laplacian operator of order $s\in (0,1)$. The associated direct problem is analyzed, as well as the inverse problem of temporal reconstruction from final data, which is inherently ill-posed. Through a spectral representation, the instability of the problem and the effect of the choice of the function $\mathrm{\Psi }$ on the amplification of spectral modes are examined.