arXiv subjects

Saumyajit Das

Publications and source records attributed to Saumyajit Das.

12 recordsLinked to original sources

arXiv2026

On Fractional Borg-Levinson Problem

In this article, we investigate the fractional Borg-Levinson problem, an inverse spectral problem focused on recovering potentials from boundary spectral data. We demonstrate that, within an admissible class of potentials, the potential is uniquely determined by the given data. However, for technical reasons, we restrict the fractional exponent to the interval $(\frac12,1)$. The admissible class is explicitly defined in our calculations and depends solely on the domain and the spatial dimension.

math.AP↗

arXiv2026

Eigenfunction equivalence for the fractional Laplace-Beltrami operator and the classical Helmholtz equation

In this article we study the spectral problem related to the fractional Laplace-Beltrami equation on $(\mathbb{R}^d,g)$ and establish its equivalence with the classical anisotropic Helmholtz equation. The proof is based on Seeley's construction of complex powers of elliptic operators and the pseudodifferential symbolic calculus. As an application, we describe the related fixed-frequency inverse scattering problem of recovering the metric from the scattering amplitude.

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arXiv2026

The Diffusive Exchange Driven Growth Model with unbounded kernels

We study the discrete diffusive exchange-driven growth (EDG) equations on a bounded smooth domain of arbitrary dimension subject to homogeneous Neumann boundary conditions. The system belongs to the class of infinite systems of semilinear partial differential equations with nonlinear source terms of quadratic type. Global-in-time existence of solutions is established for separable exchange kernels of the form $K_{i,j}=b_i a_j$, where the donor rates exhibit at most linear growth while the receiver rates are sublinear. The analysis is based on a uniform Fisher information estimate obtained from an entropy-entropy dissipation identity. This estimate yields renormalized solutions to a truncated system with finitely many species. A compactness argument then enables passage to the limit in the exchange operator, leading to the existence of global-in-time solutions for the full system.

math.AP↗

arXiv2026

Inverse scattering for the fractional Schrödinger equation

This article is devoted to studying the inverse scattering for the fractional Schrödinger equation, and in particular we solve the Born approximation problem. Based on the ($p$,$q$)-type resolvent estimate for the fractional Laplacian, we derive an expression for the scattering amplitude of the scattered solution of the fractional Schrödinger equation. We prove the uniqueness of the potential using the scattering amplitude data.

math.AP↗

arXiv2026

Homogenization of Three Species Reaction Diffusion Equation in Perforated Domains

In this article we study the asymptotic behaviour of the solution of the three species chemical reaction-diffusion model with non-homogeneous Neumann boundary condition in a perforated domain. We investigate how the mass inflow at the microscale affects the three-species reaction-diffusion system at the macroscale using two-scale convergence. As the size of the perforations vanishes, the microscale effects are captured by a global source term in the homogenized equation, which remains a three-species reaction-diffusion system but with modified diffusion coefficients.

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arXiv2026

On a nonlocal non-linear inverse scattering problem

In this article, we study the inverse scattering problem for the nonlinear fractional Helmholtz equation with cubic nonlinearity in three dimensions, where we recover a compactly supported potential from scattering amplitude.

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arXiv2026

Existence for the Discrete Nonlinear Fragmentation Equation with Degenerate Diffusion

A mathematical model for the discrete nonlinear fragmentation (collision-induced breakage) equation with diffusion is studied. The existence of global weak solutions is established in arbitrary spatial dimensions without assuming a strictly positive lower bound on the diffusion coefficients, extending previous results that were restricted to one-dimensional domains and relied on uniformly positive diffusion. The analysis is carried out under boundedness assumptions on the collision and breakage kernels. The proof is based on the construction of a suitable regularized system, combined with weak $L^2$ a priori estimates and compactness arguments in $L^1$, which allow the passage to the limit in the nonlinear fragmentation operator.

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arXiv2025

Boundary Control and Calderón type Inverse Problems in Non-local heat equation

We examine various density results related to the solutions of the non-local heat equation at a specific time slice, focusing on two distinct models: one with homogeneous Dirichlet boundary condition and the other with singular boundary data. In both the cases, we assume the non-local exponent $a\in(\frac{1}{2},1)$. We explore both the qualitative and quantitative aspects of the approximations. Additionally, we address Calderón-type inverse problems for these parabolic models, where we recover the potentials by analyzing the solutions either on the boundary or at a particular time slice. In both the density results and the Calderón type inverse problems, the Pohozaev identity plays a crucial role. Finally, in the last section, we apply the Pohozaev identity to a specific elliptic eigenvalue problem and demonstrate that the eigenfunctions, when divided by an appropriate power of the distance function, can not vanish on any non-empty open subset of the boundary. This particular eigenvalue problem does not need any restriction on the non-local exponent.

math.AP↗

arXiv2025

Anisotropic Calderón problem for a logarithmic Schrödinger operator of order $2+$ on closed Riemannian manifolds

In this article, we study the anisotropic Calderón problems for the non local logarithimic Schrödinger operators $(-Δ_g+m)\log{(-Δ_g+m)}+V$ with $m>1$ on a closed, connected, smooth Riemannian manifold of dimension $n\geq2$. We will show that, for the operator $(-Δ_g+m)\log{(-Δ_g+m)}+V$, the recovery of both the Riemannian metric and the potential is possible from the Cauchy data, in the setting of a common underlying manifold with varying metrics. This result is unconditional. The last result can be extended to the case of setwise distinct manifolds also. In particular, we demonstrate that for setwise distinct manifolds, the Cauchy data associated with the operator $(-Δ_g+m)\log{(-Δ_g+m)}+V$, measured on a suitable non-empty open subset, uniquely determines the Riemannian manifold up to isometry and the potential up to an appropriate gauge transformation. This particular result is unconditional when the potential is supported entirely within the observation set. In the more general setting-where the potential may take nonzero values outside the observation set-specific geometric assumptions are required on both the observation set and the unknown region of the manifold.

math.AP↗

arXiv2024

Convergence to equilibrium for a degenerate triangular reaction-diffusion system

In this article we study a reaction diffusion system with $m$ unknown concentration. The non-linearity in our study comes from an underlying reversible chemical reaction and triangular in nature. Our objective is to understand the large time behaviour of solution where there are degeneracies. In particular we treat those cases when one of the diffusion coefficient is zero and others are strictly positive. We prove convergence to equilibrium type of results under some condition on stoichiometric coefficients in dimension $1$,$2$ and $3$ in correspondence with the existence of classical solution. For dimension greater than 3 we prove similar result under certain closeness condition on the non-zero diffusion coefficients and with the same condition imposed on stoichiometric coefficients. All the constant occurs in the decay estimates are explicit.

math.AP↗

arXiv2024

Convergence to equilibrium for a degenerate three species reaction-diffusion system

In this work, we study a $3\times 3$ triangular reaction-diffusion system. Our main objective is to understand the long time behaviour of solutions to this reaction-diffusion system when there are degeneracies. More precisely, we treat cases when one of the diffusion coefficients vanishes while the other two diffusion coefficients stay positive. We prove convergence to equilibrium type results. In all our results, the constants appearing in the decay estimates are explicit.

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arXiv2024

Existence of solution of a triangular degenerate reaction-diffusion system

In this article we study a chemical reaction-diffusion system with $m$ unknown concentration. The non-linearity in our study comes from a particular chemical reaction where one unit of a particular species generated from other $m-1$ species and disintegrates to generate all those $m-1$ species in the same manner, i.e., triangular in nature. Our objective is to find whether global in time solution exists for this system where one or more species stops diffusing. In particular weak global in time solution exists for all the degenerate cases in any dimension. We are further able to show classical global in time solution exists for all the degenerate cases in any dimension except one and this particular case too attain classical global in time solution up to dimension $2$. We also analyze global in time existence result for the case of quadratic non-linear rate functions and also analyze a three dimensional case.

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