arXiv subjects

Nguyen Ngoc Trong

Publications and source records attributed to Nguyen Ngoc Trong.

5 recordsLinked to original sources

arXiv2026

Blow-up and blow-up-time estimates for a singular pseudo-parabolic equation with a space-time variable exponent

Let $d \in \{3,4,5,\ldots\}$ and $Ω\subset \Ri^d$ be open bounded with Lipschitz boundary. Let $Q = Ω\times (0,\infty)$ and $p \in C(\overline{Q})$ be such that \[ 2 < p^- \le p(\cdot) \le p^+ < 2^* := \frac{2d}{d-2}, \] where $ p^- := \essinf_{(x,t) \in Q} p(x,t) $ and $ p^+ := \esssup_{(x,t) \in Q} p(x,t). $ Consider the reaction-diffusion parabolic problem \[ (P) \quad \left\{\begin{array}{ll} \displaystyle\frac{u_t}{|x|^2} - Δu = k(t) \, |u|^{p(x,t)-2}u & (x,t) \in Ω\times (0,T), u(x,t) = 0, & (x,t) \in \partial Ω\times (0,T), \smallskip u(x,0) = u_0(x), & x \in Ω, \end{array}\right. \] where $T > 0$ and $0 \ne u_0 \in W^{1,2}_0(Ω)$. We investigate the existence and uniqueness of a weak solution to $(P)$. The upper and lower bounds on the blow-up time of the weak solution are also considered.

math.AP↗

arXiv2026

A Weighted Bossel-Daners Transfer Principle and a Pure-Power Robin Faber-Krahn Inequality

We prove a weighted Bossel-Daners transfer principle for the first Robin eigenvalue of the $p$-Laplacian under $w=m^{1/p'}$, where $p'=p/(p-1)$. The argument combines double-density isoperimetry with spectral admissibility for singular weights and normalized-flux monotonicity. It uses an exact $BV$ zero-extension formula, an $L^{p'}(m\,dx)$ selection lemma, and a nonatomic rank map that remains well defined on positive-measure level sets. For the singular pair \[ m_b(x)=|x|^b, \qquad w_b(x)=|x|^{b/p'}, \] known power-weight isoperimetry provides the geometric input. We establish spectral admissibility up to the critical exponent $p=N$ and, for the positive radial first eigenfunction $z=z(r)$ on the centered ball $B_R$, derive the integrated singular radial equation and center asymptotics and prove directly that \[ θ_R'(r)>0 \quad(0 0$. The range $1<p<N$ is complementary to the previously known $p\ge N$ weighted Talenti theory; at the shared endpoint $p=N$, the present proof also permits the singular contact $0\in\partialΩ$.

math.AP↗

arXiv2020

Up-to-boundary pointwise gradient estimates for very singular quasilinear elliptic equations with mixed data

This paper establishes pointwise estimates up to boundary for the gradient of weak solutions to a class of very singular quasilinear elliptic equations with mixed data: \begin{cases} -\operatorname{div}(A(x,D u))=g-\operatorname{div} f \quad & \mathrm{in} \quad Ω\\ u= 0 \quad & \text{on} \ \partial Ω, \end{cases} where $Ω\subset \mathbb{R}^n$ is sufficiently flat in the sense of Reifenberg.

math.AP↗

arXiv2020

Heat kernels of generalized degenerate Schrödinger operators and Hardy spaces

Let $\displaystyle L = -\frac{1}{w} \, \mathrm{div}(A \, \nabla u) + μ$ be the generalized degenerate Schrödinger operator in $L^2_w(\mathbb{R}^d)$ with $d\ge 3$ with suitable weight $w$ and measure $μ$. The main aim of this paper is threefold. First, we obtain an upper bound for the fundamental solution of the operator $L$. Secondly, we prove some estimates for the heat kernel of $L$ including an upper bound, the Hölder continuity and a comparison estimate. Finally, we apply the results to study the maximal function characterization for the Hardy spaces associated to the critical function generated by the operator $L$.

math.FA↗

arXiv2019

On boundedness property of singular integral operators associated to a Schrödinger operator in a generalized Morrey space and applications

In this paper, we provide the boundedness property of the Riesz transforms associated to the Schrödinger operator $\mathcal{L}=-Δ+ \mathbf{V}$ in a new weighted Morrey space which is the generalized version of many previous Morrey type spaces. The additional potential $\V$ considered in this paper is a non-negative function satisfying the suitable reverse Hölder's inequality. Our results are new and general in many cases of problems. As an application of the boundedness property of these singular integral operators, we obtain some regularity results of solutions to Schrödinger equations in the new Morrey space.

math.AP↗