Direct and inverse problems for pseudo-parabolic equations on graded Lie groups

Faculty-development competitive research grants program for 2023-2025

In this project we aim to obtain the solvability/non-solvability results to direct and inverse problems for pseudo-parabolic equations for general homogeneous invariant subelliptic differential operators on general graded Lie groups. Examples of graded Lie groups include the Euclidean group R^n, the Heisenberg group, and general stratified Lie groups. The considered class of Lie groups is the most general class of nilpotent Lie groups where one can still consider hypoelliptic homogeneous invariant differential operators and the corresponding subelliptic differential equations. The examples are the pseudo-parabolic equations for the sub-Laplacian on the Heisenberg group or on general stratified Lie groups, or p-evolution equations for higher order operators, already in all these cases our results will be new. We will also study nonlinear cases, that is, when the source function is nonlinear. In the nonlinear case, blow-up/nonexistence results will be obtained.
A connected simply connected Lie group G is called a graded (Lie) group if its Lie algebra admits a gradation. The graded groups form the subclass of homogeneous nilpotent Lie groups admitting homogeneous hypoelliptic left-invariant differential operators ([Mil80], [tER97], see also a discussion in [FR16, Section 4.1]). Such operators are called Rockland operators from the Rockland conjecture, solved by Helffer and Nourrigat [HN79]. So, we understand by a Rockland operator any left-invariant homogeneous hypoelliptic differential operator on G.
Thus, the present project is devoted to the exposition of the research developments at the intersection of two active fields of mathematics: PDEs and the theory of Lie groups. Both subjects are very broad and deserve separate studies on their own. However, a combination of the so-called subelliptic theory and our recent research techniques in the area does allow one to make a consistent treatment of pseudo-parabolic equations on graded groups, their numerous features, and a number of related topics. This brings many new insights to the subject, also allowing to underline the interesting character of its subelliptic features.
This project is an essential continuation of our previous research published as in award winning book forms [FR16] and [RS19]. This study has mainly fundamental character, makes a valuable contribution to the development of the theory of functional analysis on nilpotent Lie groups and the theory of subelliptic differential equations. Obtained results will be applied to solving various problems in mathematics and theoretical physics as well as may serve as fundaments for new university courses. Particularly, we apply obtained results to inverse problems for evolution equations (even in the Euclidean case) which may have interpretations in theoretical physics. For example, we will prove the existence and uniqueness results for both direct and inverse for the considered problems.

Expected social impacts: Most important contribution of this project to the Kazakhstani society and scientific community will be the training of three graduate students, who will get their Ph.D. and MS degrees working on this project. By conducting advanced research and publishing in top venues, they will become the next generation of academicians in Kazakhstan. Publication of fundamental results in prestigious mathematical journals and receiving citations promote raising the image of the Republic of Kazakhstan in the scientific world.