Regularization for a backward pseudo-parabolic problem involving the Caputo–Hadamard derivative

We study a backward problem for a pseudo-parabolic equation involving the Caputo–Hadamard fractional derivative, with known source term and noisy terminal observation. Using the exact logarithmic-time transformation s=log⁡(t/a), the problem is converted into an equivalent Caputo terminal-value problem, from which we derive a unique mild solution and a concise spectral representation. The initial time is included directly: under the logarithmic transformation, t = a corresponds to s = 0, so u(⋅,a) is obtained by evaluating the mild representation at the endpoint, without temporal extrapolation. Returning to the physical time variable, the solution is decomposed into a known source contribution and a terminal-data contribution governed by a bounded pseudo-parabolic multiplier. This yields a topology-dependent stability dichotomy: the terminal-to-state map is Lipschitz stable in L2(Ω), whereas the corresponding recovery in H01(Ω) is ill-posed in the sense of Hadamard. Under an H1+r(Ω)-smoothness condition, a Hölder conditional stability estimate is obtained. The dependence on the reconstruction time is retained through an explicit nonincreasing envelope Λ(t), with Λ(T)=1; the energy-space convergence rates carry the factor Λ(t)r/(r+1). We then develop Tikhonov and Landweber regularizations in the energy space and establish convergence under a priori and discrepancy-based parameter choice rules within the respective smoothness ranges. A reproducible spectral implementation, including explicit parameter-selection algorithms and a computational flowchart, is used to illustrate the stability mechanism and compare the two regularization procedures.