Research Projects

The study of the wave equations in partial differential equations (PDEs) is a fundamental aspect of mathematical analysis, with extensive applications in physics, engineering, and applied mathematics. More specifically, the analysis of long-time behavior of solutions helps in determining whether wave equation solutions increase unboundedly, decay, or remain bounded, all of which are crucial for understanding system stability. I’m also interested in studying inverse problems for the wave equation, including the spherical Radon transform, which is closely related to solutions of the wave equation. Furthermore, the study of inverse problems serves a variety of purposes, such as the reconstruction of unknown characteristics from observable data in seismology, medical imaging, photo-acoustic tomography. etc. This proposal addresses three important and interconnected problems in the context of long-time behavior and inverse problems for the wave equation. Problem 1: Long-time behavior of the wave equation associated with Schrödinger operators having continuous spectrum. In this problem we first intend to develop the decay estimates for homogeneous wave equation and then we will study the behavior of the solution to the wave equation as t → +∞. Moreover, we will also determine the Fujita exponent for the global (in-time) solution. Next, we will investigate the case of strong damping, including effective damping and non-effective damping. Problem 2: Fixed angle scattering inverse problem with point source to recover the Riemannian metric. In the Fixed angle scattering inverse problem with point source, the objective is to utilize scattered waves produced by a point source to reconstruct a medium’s properties (such as its shape, material qualities, or internal structures). In this problem, our aim is to recover the Reimannian metric using the waves generated by a point source, which is equivalent to reconstructing the shape, curvature, and geometric structure of the space. Problem 3: Explicit Inversion formula of spherical Radon transform In this problem, we first intend to look into the explicit formula for SRT in the case of even dimensions, which we think can be expressed in terms of an integral or fractional differential equation. The explicit inversion problem will be more challenging in the case of even dimensions because of technical assumptions in the Funk-Hanke’s theorem. Next, we will also look into the inversion formula for SRT on hyperbolic space Hn with partial data in the case of odd dimensions as well as even dimensions.