Carleman Inequalities for singular Parabolic Equations and Applications

الموضوعات

علوم بحتة وطبيعية

الملخص

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.

روابط وملفات

التعريف والنوع

رقم الوثيقة

0b7291c3-020f-4f67-a388-d6b8e5aff59c

رقم العقد

0

نوع الوسائط

Crawler

نوع المحتوى

الرسائل العلمية

صيغة المصدر

رسائل دكتوراة

نوع الملف

pdf text

أسماء الملفات

1109181_1.pdf, 1109181_2.pdf

بيانات النشر

ألقاب المؤلفين

[{"name_ar":"Hichem Ramoul","title_ar":"اشراف","title_en":"Supervision"},{"name_ar":"Mahdi Nedjma ","title_ar":"اعداد","title_en":"Preparation"}]

اللغة

English

المصدر والدورية

اسم المصدر

Carleman Inequalities for singular Parabolic Equations and Applications

المحتوى والصفحات

عدد الصفحات

0

إشراف وإعداد

الإشراف

Hichem Ramoul

الإعداد

Mahdi Nedjma

الاقتباسات الببليوغرافية