arXiv subjects

Uttam Kumar Dolai

Publications and source records attributed to Uttam Kumar Dolai.

4 recordsLinked to original sources

arXiv2026

Mixed Grushin Heat Equations with Riesz-Potential Nonlinearities in Marcinkiewicz Spaces: Subcritical, Critical, and Supercritical Regimes

We study the nonlinear evolution equation $$ \partial_t u+(G+G^δ)u=I_α(|u|^ρ) \quad\text{on }\mathbb{R}^{N+k}, $$ where $G$ is the nonnegative self adjoint realization of Grushin operator, $0<δ<1$, and $I_α$ is the potential operator with kernel $|z|^{-α}$, $0<α 1+p/β$, well-posedness is recovered in higher-integrability Marcinkiewicz spaces $L^{q,\infty}$ satisfying $$ β(ρ-1)<q<β_δ(ρ-1), $$ with global existence for sufficiently small initial data. In all cases, the solutions attain their initial data in the weak-$*$ sense. These results extend the Marcinkiewicz-space theory to mixed Grushin diffusion with a spatially nonlocal potential source and reveal the role of the two competing diffusion scales in determining the admissible integrability regimes.

math.AP↗

arXiv2026

Hartee-type heat equation associated to fractional anharmonic oscillator on weighted modulation spaces

We study Hartree-type nonlinear heat equations associated with fractional generalised anharmonic oscillators $A_{k,l}$ in weighted modulation spaces. We first derive Strichartz-type estimates for the associated heat semigroup and then apply them to establish global well-posedness for small initial data. For $s>\frac{d}{q'}$, the result is obtained via trilinear estimates exploiting the algebra property of the weighted modulation space, $M^{p,q}_s$. We further establish refined trilinear estimates that bypass the algebra structure, thereby extending the global well-posedness theory to the wider range $s\ge 0$.

math.AP↗

arXiv2026

Phase-Space Analysis of generalised Fractional Anharmonic and Ornstein-Uhlenbeck Semigroups on Weighted Modulation Spaces

We develop a phase-space framework for fractional generalised anharmonic oscillators and their heat semigroups on weighted modulation spaces. We consider operators of the form \[ \mathcal{H}_{k,l}=(-Δ)^{l}+V(x), \] where $V$ is a strictly positive homogeneous potential of polynomial growth of order $2k$. By studying a Hörmander metric adapted to the quasi-homogeneous symbol $|ξ|^{2l}+V(x)$, as in \cite{MR4299820, MR4944933} we place $\mathcal{H}_{k,l}$ and its fractional powers within the Weyl-Hörmander calculus. In this setting, we show that the fractional operators $\mathcal{H}_{k,l}^β$, $β>0$, are globally hypoelliptic pseudodifferential operators and derive refined symbol estimates for the heat semigroup $e^{-t\mathcal{H}_{k,l}^β}$. These estimates yield boundedness and smoothing properties of the fractional anharmonic heat semigroup on weighted modulation spaces $\mathcal{M}^{p,q}_{s}$, for the full range $0<p,q\leq\infty$ and suitable range of $s$. As applications, we establish global well-posedness of nonlinear heat equations associated with $\mathcal{H}_{k,l}^β$, including both homogenous power and spatially inhomogenous nonlinearities. Finally, we introduce Gaussian modulation spaces adapted to the Ornstein-Uhlenbeck operator and prove continuity of the corresponding semigroup, providing a phase-space perspective complementary to classical Gaussian harmonic analysis.

math.FA↗

arXiv2025

Local Dispersive and Strichartz estimates for the Schrödinger equation associated to the Ornstein-Uhlenbeck operator

In this paper we study the linear and nonlinear Schrödinger equations associated with the Ornstein-Uhlenbeck (OU) operator endowed with the Gaussian measure. While classical Strichartz estimates are well-developed for the free Schrödinger operator on Euclidean spaces, extending them to non-translation-invariant operators like the OU operator presents significant challenges due to the lack of global dispersive decay. In this work, we overcome these difficulties by deriving localized $L^1 \to L^\infty$ dispersive estimates for the OU Schrödinger propagator using Mehler kernel techniques. We then establish a family of weighted Strichartz estimates in Gaussian $L^p$ spaces via interpolation and the abstract $TT^*$-method. As an application, we prove local well-posedness results for the nonlinear Schrödinger equation with power-type nonlinearity in both subcritical and critical regimes. Our framework reveals new dispersive phenomena in the context of the OU semigroup and provides the first comprehensive Strichartz theory in this setting.

math.FA↗