arXiv subjects

Quoc Hung Phan

Publications and source records attributed to Quoc Hung Phan.

9 recordsLinked to original sources

arXiv2024

A complete description of the asymptotic behavior at infinity of positive radial solutions to $Δ^2 u = u^α$ in $\mathbf R^n$

We consider the biharmonic equation $Δ^2 u = u^α$ in $\mathbf R^n$ with $n \geqslant 1$. It was proved that this equation has a positive classical solution if, and only if, either $α\leqslant 1$ with $n \geqslant 1$ or $α\geqslant (n+4)/(n-4)$ with $n \geqslant 5$. The asymptotic behavior at infinity of all positive radial solutions was known in the case $α\geqslant (n+4)/(n-4)$ and $n \geqslant 5$. In this paper, we classify the asymptotic behavior at infinity of all positive radial solutions in the remaining case $α\leqslant 1$ with $n \geqslant 1$; hence obtaining a complete picture of the asymptotic behavior at infinity of positive radial solutions. Since the underlying equation is higher order, we propose a new approach which relies on a representation formula and asymptotic analysis arguments. We believe that the approach introduced here can be conveniently applied to study other problems with higher order operators.

math.AP↗

arXiv2019

Exhaustive existence and non-existence results for some prototype polyharmonic equations in the whole space

In this paper, we are interested in entire, non-trivial, non-negative solutions and/or entire, positive solutions to the simplest models of polyharmonic equations with power-type nonlinearity \[ Δ^m u = \pm u^α \quad \text{ in } \mathbb R^n \] with $n \geqslant 1$, $m \geqslant 1$, and $α\in \mathbb R$. We aim to study the existence and non-existence of such classical solutions to the above equations in the full range of the constants $n$, $m$ and $α$. Remarkably, we are able to provide necessary and sufficient conditions on the exponent $α$ to guarantee the existence of such solutions in $\mathbb R^n$. Finally, we identify all the situations where any entire non-trivial, non-negative classical solution must be positive.

math.AP↗

arXiv2018

Higher order Sobolev trace inequalities on balls revisited

Inspired by a recent sharp Sobolev trace inequality of order four on the balls $\mathbb B^{n+1}$ found by Ache and Chang [AC15], we propose a slightly different approach to reprove Ache-Chang's trace inequality. To illustrate this approach, we reprove the classical Sobolev trace inequality of order two on $\mathbb B^{n+1}$ and provide sharp Sobolev trace inequalities of orders six and eight on $\mathbb B^{n+1}$. As the limiting case of the Sobolev trace inequality, a Lebedev-Milin type inequality of order up to eight is also considered.

math.AP↗

arXiv2017

A pointwise inequality for a biharmonic equation with negative exponent and related problems

Inspired by a recent pointwise differential inequality for positive bounded solutions of the fourth-order Hénon equation $Δ^2 u = |x|^a u^p$ in ${\mathbb R}^n$ with $a \geqslant 0$, $p > 1$, $n \geqslant 5$ due to Fazly, Wei, and Xu [ Anal. PDE., 8(2015) 1541--1563], first for some positive constants $α$ and $β$ we establish the following pointwise inequality \[ Δu \geqslant αu^{-\frac{q-1}2} + βu^{-1} |\nabla u|^2 \] in ${\mathbb R}^n$ with $n \geqslant 3$ for positive $C^4$-solutions of the fourth-order equation \[ Δ^2u=-u^{-q} \quad \text{ in } \mathbb R^n \] where $q > 1$. Next, we prove a comparison property for Lane--Emden system with exponents of mixed sign. Finally, we give an analogue result for parabolic models by establishing a comparison property for parabolic system of Lane--Emden type. To obtain all these results, a new argument of maximum principle is introduced, which allows us to deal with solutions with high growth at infinity. We expect to see more applications of this new method to other problems in different contexts.

math.AP↗

arXiv2015

A Liouville-type theorem for the $3$-dimensional parabolic Gross-Pitaevskii and related systems

We prove a Liouville-type theorem for semilinear parabolic systems of the form $${\partial_t u_i}-Δu_i =\sum_{j=1}^{m}β_{ij} u_i^ru_j^{r+1}, \quad i=1,2,...,m$$ in the whole space ${\mathbb R}^N\times {\mathbb R}$. Very recently, Quittner [{\em Math. Ann.}, DOI 10.1007/s00208-015-1219-7 (2015)] has established an optimal result for $m=2$ in dimension $N\leq 2$, and partial results in higher dimensions in the range $p< N/(N-2)$. By nontrivial modifications of the techniques of Gidas and Spruck and of Bidaut-Véron, we partially improve the results of Quittner in dimensions $N\geq 3$. In particular, our results solve the important case of the parabolic Gross-Pitaevskii system -- i.e. the cubic case $r=1$ -- in space dimension $N=3$, for any symmetric $(m,m)$-matrix $(β_{ij})$ with nonnegative entries, positive on the diagonal. By moving plane and monotonicity arguments, that we actually develop for more general cooperative systems, we then deduce a Liouville-type theorem in the half-space ${\mathbb R}^N_+\times {\mathbb R}$. As applications, we give results on universal singularity estimates, universal bounds for global solutions, and blow-up rate estimates for the corresponding initial value problem.

math.AP↗

arXiv2015

Liouville-type theorems for polyharmonic Hénon-Lane-Emden system

We study Liouville-type theorem for polyharmonic Hénon-Lane-Emden system $(-Δ)^mu=|x|^av^p,\; (-Δ)^mv=|x|^bu^q$ when $m,p,q\geq 1, pq\ne 1$, and $a,b\geq 0$. It is a natural conjecture that the nonexistence of positive solutions should be true if and only if $(N+a)/(p+1)+$ $(N+b)/(q+1)>N-2m$. It is shown by Fazly [6] that the conjecture holds for radial solutions in all dimensions and for classical solutions in dimension $N\leq 2m+1$. We here give some partial results in dimension $N\geq 2m+2$.

math.AP↗

arXiv2012

Singularity and blow-up estimates via Liouville-type theorems for Hardy-Hénon parabolic equations

We consider the Hardy-Hénon parabolic equation $u_t-Δu =|x|^a |u|^{p-1}u$ with $p>1$ and $a\in {\mathbb R}$. We establish the space-time singularity and decay estimates, and Liouville-type theorems for radial and nonradial solutions. As applications, we study universal and a priori bound of global solutions as well as the blow-up estimates for the corresponding initial boundary value problem.

math.AP↗

arXiv2012

Global existence of solutions for a chemotaxis-type system arising in crime modeling

We consider a nonlinear, strongly coupled, parabolic system arising in the modeling of burglary in residential areas. The system is of chemotaxis-type and involves a logarithmic sensivity function and specific interaction and relaxation terms. Under suitable assumptions on the data of the problem, we give a rigorous proof of the existence of a global and bounded, classical solution, thereby solving a problem left open in previous work on this model. Our proofs are based on the construction of approximate entropies and on the use of various functional inequalities. We also provide explicit numerical conditions for global existence when the domain in a square, including concrete cases involving values of the parameters which are expected to be physically relevant

math.AP↗

arXiv2011

Liouville-type theorems and bounds of solutions for Hardy-Hénon elliptic systems

We consider the Hardy-Hénon system $-Δu =|x|^a v^p$, $-Δv =|x|^b u^q$ with $p,q>0$ and $a,b\in {\mathbb R}$ and we are concerned in particular with the Liouville property, i.e. the nonexistence of positive solutions in the whole space ${\mathbb R}^N$. In view of known results, it is a natural conjecture that this property should be true if and only if $(N+a)/(p+1)+$ $(N+b)/(q+1)>N-2$. In this paper, we prove the conjecture for dimension N=3 in the case of bounded solutions and in dimensions $N\le 4$ when $a,b\le 0$, among other partial nonexistence results. As far as we know, this is the first optimal Liouville type result for the Hardy-Hénon system. Next, as applications, we give results on singularity and decay estimates as well as a priori bounds of positive solutions.

math.AP↗