arXiv subjects

Abhilash Tushir

Publications and source records attributed to Abhilash Tushir.

11 recordsLinked to original sources

arXiv2026

A constructive approach to range characterization of spherical mean transform in odd dimensions

This work focuses on range characterization of spherical mean transform (SMT) in odd-dimension Euclidean space. In a recent work, a simpler description characterizing the range of SMT in odd dimensions was derived. This involved symmetry relations involving certain linear ordinary differential operators acting on the coefficients of the spherical harmonics expansion of the function in the range. The sufficiency part of this characterization was shown using an existing range characterization result due to Agranovsky-Kuchment-Quinto. In the current work, we provide a direct and constructive proof showing the sufficiency of the aforementioned symmetry conditions and we obtain this without invoking any of the existing range characterization results. As an added advantage, given a function $g$ satisfying the range conditions, our approach provides an explicit construction of a function $f$ such that the SMT of $f$ is $g$.

math.AP↗

arXiv2026

A Unified Range Characterization for the Spherical mean transform

In this article, we investigate the range characterization for the spherical mean transform (SMT) of functions supported in the unit ball. In earlier works, in the case of odd dimensions, a set of differential conditions was obtained, whereas in the case of even dimensions, integral conditions were obtained. We prove that these conditions that are different based on the parity of dimension are, in fact, equivalent in odd dimensions. This equivalence shows that the integral conditions yield a unified simple range characterization for the SMT that is valid in both even and odd dimensions.

math.AP↗

arXiv2026

Explicit inversion of spherical Radon transforms in odd dimensions with partial radial data

We derive an explicit inversion algorithm for the spherical Radon transform in odd dimensions with partial radial data. We prove that the reconstruction of the unknown function can be reduced to solving ordinary differential equations, thereby providing a more explicit approach in odd dimensions than solving Volterra integral equation of the first kind established in prior works. We also provide analytical solutions in some special cases. Finally, we present numerical simulations validating our theoretical results. Our work answers a question posed by Rubin in ``Inversion formulae for the spherical mean in odd dimensions and the Euler-Poisson-Darboux equation,'' Inverse Problems 24 (2008), no. 2, 025021, 10 pp.

math.AP↗

arXiv2025

Structurally damped semilinear evolution equation for positive operators on Hilbert space

In this study, we analyze a semilinear damped evolution equation under different damping conditions, including the undamped $(θ=0)$, effectively damped $(0<2θ<σ)$, critically damped $(2θ=σ)$, and non-effectively damped $(σ<2θ\leq 2σ)$. The analysis is conducted in two parts; the present article is devoted to examining decay estimates of solutions to the linear evolution equation governed by a self-adjoint, positive operator $\mathcal{L}$ with discrete spectrum subject to initial Cauchy data of minimal regularity. Specifically, we consider the Cauchy problem: \begin{equation*} \left\{\begin{array}{l} u_{tt}(t)+\mathcal{L}^θu_{t}(t)+\mathcal{L}^σu(t) =0, \quad t>0, u(0)=u_{0}\in\mathcal{H},\quad u_{t}(0)=u_{1}\in\mathcal{H}, \end{array}\right. \end{equation*} in different damping conditions. %More precisely, we study decay estimates for a solution, its time derivative, and space derivative in both cases. Furthermore, we demonstrate that the decay rates of the associated solutions improve with the regularity of the initial Cauchy data. As an application of the decay estimates, we also demonstrate the global existence (in time) of the solution in certain cases, taking into account polynomial-type nonlinearity. In the 2nd article, we will address the remaining instances where global existence cannot be assured and instead present findings on local existence and possible blow-up results.

math.AP↗

arXiv2025

Fractional semilinear damped wave equation on the Heisenberg group

This paper aims to investigate the Cauchy problem for the semilinear damped wave equation for the fractional sub-Laplacian $(-\mathcal{L}_{\mathbb{H}})^α$, $α>0$ on the Heisenberg group $\mathbb{H}^{n}$ with power type non-linearity. With the presence of a positive damping term and nonnegative mass term, we derive $L^2-L^2$ decay estimates for the solution of the homogeneous linear fractional damped wave equation on $\mathbb{H}^{n}$, for its time derivative, and for its space derivatives. We also discuss how these estimates can be improved when we consider additional $L^1$-regularity for the Cauchy data in the absence of the mass term. Also, in the absence of mass term, we prove the global well-posedness for $2\leq p\leq 1+\frac{2α}{(\mathcal{Q}-2α)_{+}}$ $(\text{or }1+\frac{4α}{\mathcal{Q}}<p\leq 1+\frac{2α}{(\mathcal{Q}-2α)_{+}})$ in the case of $L^1\cap L^2$ $(\text{or } L^2)$ Cauchy data, respectively. However, in the presence of the mass term, the global (in time) well-posedness for small data holds for $1<p \leq 1+ \frac{2α}{(\mathcal{Q}-2α)_{+}}$. Finally, as an application of the linear decay estimates, we investigate well-posedness for the Cauchy problem for a weakly coupled system with two semilinear fractional damped wave equations with positive mass term on $\mathbb{H}^{n}$.

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arXiv2024

Global well-posedness of space-time fractional diffusion equation with Rockland operator on graded Lie group

In this article, we examine the general space-time fractional diffusion equation for left-invariant hypoelliptic homogeneous operators on graded Lie groups. Our study covers important examples such as the time-fractional diffusion equation, the space-time fractional diffusion equation when diffusion is under the influence of sub-Laplacian on the Heisenberg group, or general stratified Lie groups. We establish the global well-posedness of the Cauchy problem for the general space-time fractional diffusion equation for the Rockland operator on a graded Lie group in the associated Sobolev spaces. More precisely, we establish the existence and uniqueness results for both homogeneous and inhomogeneous fractional diffusion equations. In addition, we also develop some regularity estimates.

math.AP↗

arXiv2024

Time-fractional discrete diffusion equation for Schrödinger operator

This article aims to investigate the semi-classical analog of the general Caputo-type diffusion equation with time-dependent diffusion coefficient associated with the discrete Schrödinger operator, $\mathcal{H}_{\hbar,V}:=-\hbar^{-2}\mathcal{L}_{\hbar}+V$ on the lattice $\hbar\mathbb{Z}^{n},$ where $V$ is a non-negative multiplication operator and $\mathcal{L}_{\hbar}$ is the discrete Laplacian. We establish the well-posedness of the Cauchy problem for the general Caputo-type diffusion equation with a regular coefficient in the associated Sobolev-type spaces. However, it is very weakly well-posed when the diffusion coefficient has a distributional singularity. Finally, we recapture the classical solution (resp. very weak) for the general Caputo-type diffusion equation in the semi-classical limit $\hbar\to 0$.

math.AP↗

arXiv2023

Discrete time-dependent wave equation for the Schrödinger operator with unbounded potential

In this article, we investigate the semiclassical version of the wave equation for the discrete Schrödinger operator, $\mathcal{H}_{\hbar,V}:=-\hbar^{-2}\mathcal{L}_{\hbar}+V$ on the lattice $\hbar\mathbb{Z}^{n},$ where $\mathcal{L}_{\hbar}$ is the discrete Laplacian, and $V$ is a non-negative multiplication operator. We prove that $\mathcal{H}_{\hbar,V}$ has a purely discrete spectrum when the potential $V$ satisfies the condition $|V(k)|\to \infty$ as $|k|\to\infty$. We also show that the Cauchy problem with regular coefficients is well-posed in the associated Sobolev type spaces and very weakly well-posed for distributional coefficients. Finally, we recover the classical solution as well as the very weak solution in certain Sobolev type spaces as the limit of the semiclassical parameter $\hbar\to 0$.

math.AP↗

arXiv2023

Discrete Heat Equation with irregular thermal conductivity and tempered distributional data

In this paper, we consider a semi-classical version of the nonhomogeneous heat equation with singular time-dependent coefficients on the lattice $\hbar \mathbb{Z}^n$. We establish the well-posedeness of such Cauchy equations in the classical sense when regular coefficients are considered, and analyse how the notion of very weak solution adapts in such equations when distributional coefficients are regarded. We prove the well-posedness of both the classical and the very weak solution in the weighted spaces $\ell^{2}_{s}(\hbar \mathbb{Z}^n)$, $s \in \mathbb{R}$, which is enough to prove the well-posedness in the space of tempered distributions $\mathcal{S}'(\hbar \mathbb{Z}^n)$. Notably, when $s=0$, we show that for $\hbar \rightarrow 0$, the classical (resp. very weak) solution of the heat equation in the Euclidean setting $\mathbb{R}^n$ is recaptured by the classical (resp. very weak) solution of it in the semi-classical setting $\hbar \mathbb{Z}^n$.

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arXiv2022

Discrete Time-Dependent Wave Equations II. Semiclassical Fractional Klein-Gordon Equation

In this paper, we consider a semiclassical version of the fractional Klein-Gordon equation on the lattice, $h{\mathbb{Z}}^n.$ Contrary to the Euclidean case that was considered in [2], the discrete fractional Klein-Gordon equation is well-posed in $\ell^2(h{\mathbb{Z}}^n).$ However, we also recover the well-posedness results in the certain Sobolev spaces in the limit of the semiclassical parameter $h\rightarrow 0.$

math.AP↗

arXiv2021

Discrete time-dependent wave equations I. Semiclassical analysis

In this paper we consider a semiclassical version of the wave equations with singular Hölder time-dependent propagation speeds on the lattice $\hbar\mathbb{Z}^{n}$. We allow the propagation speed to vanish leading to the weakly hyperbolic nature of the equations. Curiously, very much contrary to the Euclidean case considered by Colombini, de Giorgi and Spagnolo [2] and by other authors, the Cauchy problem, in this case, is well-posed in $\ell^2(\hbar\mathbb{Z}^{n})$. However, we also recover the well-posedness results in the intersection of certain Gevrey and Sobolev spaces in the limit of the semiclassical parameter $\hbar\to 0$.

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