| Abstract: |
| We study backward inverse problems for coupled Cahn--Hilliard--reaction--diffusion systems. Our main result is a Carleman estimate for a fourth-order/second-order parabolic system with cross-diffusion terms. As a consequence, we derive conditional stability estimates for the reconstruction of past states from a single final-time observation: H\older stability at positive times and logarithmic stability for the initial datum. We then apply the abstract theory to a phase-field tumour growth model coupling the tumour volume fraction with the nutrient concentration. In this setting, we obtain backward uniqueness and quantitative stability for the recovery of early tumour states, extending previous results and removing additional restrictions on the chemotactic coupling. We also discuss how these results support Lipschitz stability on finite-dimensional admissible sets, which is relevant for reconstruction algorithms. |
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