Searcharxiv⌕ Search

arXiv subjects

Le Xuan Truong

Publications and source records attributed to Le Xuan Truong.

10 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

Higher Regularity of Homogeneous Gradient Compositions for $p$-Laplace-Type Equations

In this paper, we study higher regularity of homogeneous functions of the gradient of solutions to the inhomogeneous $p$-Laplace equation $\operatorname{div}(|Du|^{p-2}Du)=f$. Although a solution need not be of class $C^2$ across its critical set, its gradient is locally Hölder continuous. Suppose that $Du\in C^{0,α}_{\rm loc}$ with $α\le 1/(p-1)$, and let $Φ$ be smooth away from the origin and positively homogeneous of degree $m$. We prove that $Φ(Du)\in C^k_{\rm loc}$ whenever $m>k/α$. Moreover, all its derivatives of order at most $k$ vanish on the critical set. The proof uses the intrinsic scale $r\simeq |Du|^{1/α}$, Schauder estimates for a normalized uniformly elliptic equation, and an extension lemma across the critical set. We also obtain corresponding results for autonomous anisotropic equations and for elliptic and parabolic $p$-Laplace systems, under the appropriate Hölder assumption on the gradient. Finally, the same argument gives $C^k$ regularity criteria for high powers of nonnegative solutions to the porous medium equation.

math.AP↗
arXiv2021

Hölder continuity of solutions for a class of drift-diffusion equations

We provide several regularity results for non-homogeneous drift-diffusion equations with applications to general dissipative SQG. Our results unify in a rather simple way several previously known results. We build the estimates on an algebraic identity (for the refined energy argument) which relates any integral operator with pure powers of the laplacian.

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↗
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↗
arXiv2011

Large time behavior of differential equations with drifted periodic coefficients modeling Carbon storage in soil

This paper is concerned with the linear ODE in the form $y'(t)=λρ(t)y(t)+b(t)$, $λ<0$ which represents a simplified storage model of the carbon in the soil. In the first part, we show that, for a periodic function $ρ(t)$, a linear drift in the coefficient $b(t)$ involves a linear drift for the solution of this ODE. In the second part, we extend the previous results to a classical heat non-homogeneous equation. The connection with an analytic semi-group associated to the ODE equation is considered in the third part. Numerical examples are given.

math.AP↗
arXiv2011

Existence, blow-up and exponential decay estimates for a nonlinear wave equation with boundary conditions of two-point type

This paper is devoted to study a nonlinear wave equation with boundary conditions of two-point type. First, we state two local existence theorems and under suitable conditions, we prove that any weak solutions with negative initial energy will blow up in finite time. Next, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions. Finally, we present numerical results

math.AP↗
arXiv2007

Existence and Decay of Solutions of a Nonlinear Viscoelastic Problem with a Mixed Nonhomogeneous Condition

We study the initial-boundary value problem for a nonlinear wave equation given by u_{tt}-u_{xx}+\int_{0}^{t}k(t-s)u_{xx}(s)ds+ u_{t}^{q-2}u_{t}=f(x,t,u) , 0 < x < 1, 0 < t < T, u_{x}(0,t)=u(0,t), u_{x}(1,t)+ηu(1,t)=g(t), u(x,0)=û_{0}(x), u_{t}(x,0)={û}_{1}(x), where η\geq 0, q\geq 2 are given constants {û}_{0}, {û}_{1}, g, k, f are given functions. In part I under a certain local Lipschitzian condition on f, a global existence and uniqueness theorem is proved. The proof is based on the paper [10] associated to a contraction mapping theorem and standard arguments of density. In Part} 2, under more restrictive conditions it is proved that the solution u(t) and its derivative u_{x}(t) decay exponentially to 0 as t tends to infinity.

math.AP↗