Carleman Inequalities for singular Parabolic Equations and Applications
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2026
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Abstract
This thesis is devoted to the study of inverse source problems for singular and
degenerate parabolic equations, including time-fractional diffusion models.
Such problems arise naturally in many applications, but they are intrinsically
ill-posed due to the smoothing effects of diffusion and, in the present setting,
the presence of inverse-square singular potentials. These singularities destroy
uniform ellipticity and require refined analytical tools to recover stability and
uniqueness.
The first part of the thesis adopts an optimal control perspective to address
inverse source identification problems. The unknown source is treated
as a control variable and recovered as the minimizer of a suitable cost functional
involving final-time observations. Under subcritical assumptions on
the singular potential, we establish the well-posedness of the direct problem
in appropriate weighted Sobolev spaces, prove the existence of optimal
controls, and derive stability estimates for the inverse problem. Numerical
reconstruction algorithms based on iterative regularization methods are
proposed and validated through computational experiments.
The analysis is then extended to time-fractional diffusion equations with
inverse-square potentials. In this setting, the nonlocal nature of fractional
time derivatives introduces additional analytical and numerical challenges.
Nevertheless, uniqueness results for the inverse source problem are obtained,
and numerical simulations illustrate the influence of the fractional order and
noise on the reconstruction process.
The final part of the thesis is devoted to the study of Carleman inequalities
for singular parabolic operators. We develop a shifted Carleman framework
adapted to operators with interior inverse-square singularities and establish
several key intermediate estimates. While the complete global Carleman
inequality is not yet achieved, the analytical structure is clearly identified,
and the main technical difficulties—particularly those related to localization
arguments and Caccioppoli-type inequalities with singular weights—are
thoroughly analyzed. This work lays the foundation for ongoing research
conducted in collaboration with the supervisor.