arXiv subjects

Saikatul Haque

Publications and source records attributed to Saikatul Haque.

12 recordsLinked to original sources

arXiv2026

On the Hardy-Hénon heat equation with an inverse square potential

We study Cauchy problem for the Hardy-Hénon parabolic equation with an inverse square potential, namely, \[\partial_tu -Δu+a|x|^{-2} u= |x|^γ F_α(u),\] where $a\ge-(\frac{d-2}{2})^2,$ $γ\in \mathbb R$, $α>1$ and $F_α(u)=μ|u|^{α-1}u, μ|u|^α$ or $μu^α$, $μ\in \{-1,0,1\}$. We establish sharp fixed time-time decay estimates for heat semigroups $e^{-t (-Δ+ a|x|^{-2})}$ in weighted Lebesgue spaces. This may be of independent interest. As an application, we establish local well-posedness in scale subcritical and critical weighted Lebesgue spaces and small data global existence in critical weighted Lebesgue spaces. Further, under certain conditions on $γ$ and $α,$ we show that local solution cannot be extended to global one for certain initial data in the subcritical regime. Thus, finite time blow-up in the subcritical Lebesgue space norm is exhibited. We also demonstrate nonexistence of local positive weak solution (and hence failure of local well-posedness) in supercritical case for $α>1+\frac{2+γ}{d}$ the Fujita exponent.

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arXiv2025

Growth of Fourier--Lebesgue norms for mKdV

We demonstrate inflation of Fourier--Lebesgue norms for solutions to the focusing modified Korteweg--de Vries equation posed on the real line. For $p\neq 2$ and all $s\in \mathbb{R}$, we construct a sequence of solutions $u_n$ whose initial data $u_n(0)$ converges to zero in the Fourier--Lebesgue spaces $\mathcal F L^p_s(\mathbb{R})$, but whose evolutions at later times $t_n$ diverge to infinity.

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arXiv2025

The mixed fractional Hartree equations in Fourier amalgam and modulation spaces

We prove local and global well-posedness for mixed fractional Hartree equation and with low regularity Cauchy data in Fourier amalgam $\F W(L^p,\ell^q)$ and modulation $M^{p,q}$ spaces. Similar results also hold for the Hartree equation with harmonic potential in some modulation spaces. Our approach also addresses Hartree-Fock equations of finitely many (but arbitrary large) particles. A key ingredient of our method is to establish trilinear estimates for Hartree non-linearity and the use of Strichartz estimates. As a consequence, we could gain $\F W(L^p,\ell^q)$ and $M^{p,q}-$regularity for all $p,q\in [1, \infty].$ In particular, we extend result of Bhimani-Grillakis-Okoudju \cite{bhimani2020hartree} in $M^{p,q}$ for all $p,q$ and complement known results in Sobolev spaces.

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arXiv2024

Global well-posedness and equicontinuity for mKdV in modulation spaces

We establish global well-posedness for both the defocusing and focusing complex-valued modified Korteweg--de Vries equations on the real line in modulation spaces $M_p^{s,2}(\mathbb{R})$, for all $1\leq p<\infty$ and $0\leq s<3/2-1/p$. We will also show that such solutions admit global-in-time bounds in these spaces and that equicontinuous sets of initial data lead to equicontinuous ensembles of orbits. Indeed, such information forms a crucial part of our well-posedness argument.

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arXiv2023

Strong ill-posedness for fractional Hartree and cubic NLS Equations

We consider fractional Hartree and cubic nonlinear Schrödinger equations on Euclidean space $\mathbb R^d$ and on torus $\mathbb T^d$. We establish norm inflation (a stronger phenomena than standard ill-posedness) at every initial data in Fourier amalgam spaces with negative regularity. In particular, these spaces include Fourier-Lebesgue, modulation and Sobolev spaces. We further show that this can be even worse by exhibiting norm inflation with an infinite loss of regularity. To establish these phenomena, we employ a Fourier analytic approach and introduce new resonant sets corresponding to the fractional dispersion $(-Δ)^{α/2}$. In particular, when dispersion index $α$ is large enough, we obtain norm inflation {above} scaling critical regularity in some of these spaces. It turns out that our approach could treat both equations (Hartree and power-type NLS) in a unified manner. The method should also work for a broader range of nonlinear equations with Hartree-type nonlinearity.

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arXiv2023

Remark on the ill-posedness for KdV-Burgers equation in Fourier amalgam spaces

We have established (a weak form of) ill-posedness for the KdV-Burgers equation on a real line in Fourier amalgam spaces $\widehat{w}_s^{p,q}$ with $s<-1$. The particular case $p=q=2$ recovers the result of L. Molinet and F. Ribaud [Int. Math. Res. Not., (2002), pp. 1979-2005]. The result is new even in Fourier Lebesgue space $\mathcal{F}L_s^q$ which corresponds to the case $p=q(\neq 2)$ and in modulation space $M_s^{2,q}$ which corresponds to the case $p=2,q\neq 2$.

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arXiv2022

On inhomogeneous heat equation with inverse square potential

We study inhomogeneous heat equation with inverse square potential, namely, \[\partial_tu + \mathcal{L}_a u= \pm |\cdot|^{-b} |u|^αu,\] where $\mathcal{L}_a=-Δ+ a |x|^{-2}.$ We establish some fixed-time decay estimate for $e^{-t\mathcal{L}_a}$ associated with inhomogeneous nonlinearity $|\cdot|^{-b}$ in Lebesgue spaces. We then develop local theory in $L^q-$ scaling critical and super-critical regime and small data global well-posedness in critical Lebegue spaces. We further study asymptotic behaviour of global solutions by using self-similar solutions, provided the initial data satisfies certain bounds. Our method of proof is inspired from the work of Slimene-Tayachi-Weissler (2017) where they considered the classical case, i.e. $a=0$.

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arXiv2022

The Hartree and Hartree-Fock equations in Lebesgue $L^p$ and Fourier-Lebesgue $\hat{L}^p$ spaces

We establish some local and global well-posedness for Hartree-Fock equations of $N$ particles (HFP) with Cauchy data in Lebesgue spaces $L^p \cap L^2 $ for $1\leq p \leq \infty$. Similar results are proven for fractional HFP in Fourier-Lebesgue spaces $ \hat{L}^p \cap L^2 \ (1\leq p \leq \infty).$ On the other hand, we show that the Cauchy problem for HFP is ill-posed if we simply work in $\hat{L}^p \ (2<p\leq \infty).$ Analogue results hold for reduced HFP. In the process, we prove the boundedeness of various trilinear estimates for Hartree type non linearity in these spaces which may be of independent interest. As a consequence, we get natural $L^p$ and $\hat{L}^p$ extension of classical well-posedness theories of Hartree and Hartree-Fock equations with Cauchy data in just $L^2-$based Sobolev spaces.

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arXiv2021

Strichartz Estimates for Schr{ö}dinger equation with singular and time dependent Potential and Application to NLS

We establish inhomogeneous Strichartz Estimates for the Schr{ö}dinger equation with singular and time dependent potentials for non-admissible pairs. Our work extends the results provided by Vilela [23] and Foschi [6] where they proved the results in the absence of potential. It also extends the works of Pierfelice [20] and Burq, Planchon, Stalker, Tahvildar-Zadeh [3], who proved the estimates for admissible pairs. We also extend the recent work of Mizutani, Zhang, Zheng [17] and as an application of it, we improve the stability result of Kenig-Merle [13], which in turn establishes a proof (alternative to [26]) of existence of scattering solution for the energy critical focusing NLS with inverse square potential.

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arXiv2021

Norm inflation for BBM equation in Fourier amalgam and Wiener amalgam spaces with negative regularity

We consider Benjamin-Bona-Mahony (BBM) equation of the form $$ u_t+u_x+uu_x-u_{xxt}=0, \quad (x, t)\in \mathcal{M}\times \mathbb R $$ where $\mathcal{M}= \mathbb T$ or $\mathbb R.$ We establish norm inflation (NI) with infinite loss of regularity at general initial data in Fourier amalgam and Wiener amalgam spaces with negative regularity. This strengthen several known NI results at zero initial data in $H^s(\mathbb T)$ established by Bona-Dai (2017) and ill-posedness result established by Bona-Tzvetkov (2008) and Panthee (2011) in $H^s(\mathbb R).$ Our result is sharp with respect to local well-posedness result of Banquet-Villamizar-Roa (2021) in modulation spaces $M^{2,1}_s(\mathbb R)$ for $s\geq 0$.

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arXiv2021

Norm inflation with infinite loss of regularity at general initial data for nonlinear wave equations in Wiener amalgam and Fourier amalgam spaces

We study the strong ill-posedness (norm inflation with infinite loss of regularity) for the nonlinear wave equation at every initial data in Wiener amalgam and Fourier amalgam spaces with negative regularity. In particular these spaces contain Fourier-Lebesgue, Sobolev and some modulation spaces. The equations are posed on $\mathbb R^d$ and on torus $\mathbb T^d$ and involve a smooth power nonlinearity. Our results are sharp with respect to well-posedness results of Bényi and Okoudjou (2009) and Cordero and Nicola (2009) in the Wiener amalgam and modulation space cases. In particular, we also complement norm inflation result of Christ, Colliander and Tao (2003) and Forlano and Okamoto (2020) by establishing infinite loss of regularity in the aforesaid spaces.

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arXiv2018

A note on the optimal boundary regularity for the planar generalized $p$-Poisson equation

In this note, we establish sharp regularity for solutions to the following generalized $p$- Poisson equation $$-\ div\ \big(\langle A\nabla u,\nabla u\rangle^{\frac{p-2}{2}}A\nabla u\big)=-\ div\ \mathbf{h}+f$$ in the plane (i.e. in $\mathbb{R}^n=\mathbb{R}^2$) for $p>2$ in the presence of Dirichlet as well as Neumann boundary conditions and with $\mathbf{h}\in C^{1-n/q}$, $f\in L^q$, $2=n<q\leq\infty$. The regularity assumptions on the principal part $A$ as well as that on the Dirichlet/Neumann conditions are exactly the same as in the linear case and therefore sharp (see Remark 2.5 below). Our main results Theorem 2.3 and Theorem 2.4 should be thought of as the boundary analogues of the sharp interior regularity result established in the recent interesting paper [1] in the case of \begin{equation}\label{e0} -\ div\ (|\nabla u|^{p-2} \nabla u) =f \end{equation} for more general variable coefficient operators and with an additional divergence term.

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