arXiv subjects

Pengxiu Yu

Publications and source records attributed to Pengxiu Yu.

6 recordsLinked to original sources

arXiv2026

Sign-preserving solutions to the Tzitz\'eica equation on lattice graphs

On the lattice graph $\mathbb{Z}^n$, we establish the existence of sign-preserving solutions to the Tzitz\'eica equation. We prove the existence of positive solutions and two classes of negative solutions under different assumptions, and derive their decay estimates. The proof is based on a suitable approximation scheme, the monotone convergence theorem, and an exhaustion argument. These results extend those of Hua, Huang, and Wang (Anal. PDE, 2026) by establishing the existence of both positive and negative sign-preserving solutions together with their decay estimates.

math.AP↗

arXiv2026

The lifespan of positive solutions of heat equation with power-logarithmic nonlinearity on locally finite graph

On a locally finite connected graph $G=(V,E)$, using the first eigenvalue method introduced by Kaplan \cite{MR160044} and the discrete Phragm\'{e}n-Lindel\"{o}f principle developed by Hu-Wang \cite{cvhuyuanyang}, we first establish the asymptotic behaviour of the lifespan of positive solutions to a semilinear heat equation with the power-logarithmic nonlinearity $u^p|\log u|^q$, provided that the initial datum is bounded below by a positive constant. These results extend those of Hu-Wang \cite{cvhuyuanyang} to equations with a power-logarithmic source term. Moreover, by means of a more direct argument, we show that analogous lifespan estimates remain valid for nonnegative initial datum $u(x,0)$, provided that $u(x_i,0)$ is suitably large at some vertex $x_i\in V$.

math.AP↗

arXiv2025

Decay Properties of Invariant Measure and Application to Elliptic Homogenization of Non-divergence Form with an Interface

Using the self-contained PDE analysis, this paper investigates the existence and the decay properties of the invariant measure in elliptic homogenization of non-divergence form with an interface assumptions on the leading coefficient $A$ and the drift $b$ for $b_1\equiv 0$, which partially provides an alternative proof of the previous work by Hairer and Manson [Ann. Probab. 39(2011) 648-682]. Moreover, as a direct application after using the analysis by the second author [Calc. Var. Partial Differ. Equ. 64(2025) No. 114], we obtain the quantitative estimates for the homogenization problem.

math.AP↗

arXiv2024

Large-scale boundary estimates of parabolic homogenization over rough boundaries

In this paper, for a family of second-order parabolic system or equation with rapidly oscillating and time-dependent periodic coefficients over rough boundaries, we obtain the large-scale boundary estimates, by a quantitative approach. The quantitative approach relies on approximating twice: we first approximate the original parabolic problem over rough boundary by the same equation over a non-oscillating boundary and then approximate the oscillating equation over a non-oscillating boundary by its homogenized equation over the same non-oscillating boundary.

math.AP↗

arXiv2024

Existence for nonlinear fractional p-Laplacian equations on finite graphs

In this paper, we assume that $q>0$, $p>1$ and $s\in(0,1)$ , and consider the following nonlinear fractional p-Laplacian equations on finite graphs: \begin{equation*} \left\{ \begin{array}{lll} \partial_t u^q(x,t)+(-\Delta)_p^su=0,\\[15pt] u(x,t)|_{t=0}=u_0>0, \end{array} \right. \end{equation*} where $(-\Delta)_p^s$ is fractional Laplace operator on finite graphs. We establish the existence of solutions to the above parabolic equation using an iterative approach, which is different from previous works on graphs. Furthermore, we also derive some energy estimate of the solution.

math.AP↗

arXiv2024

Existence results for Kazdan-Warner type equations on graphs

In this paper, motivated by the work of Huang-Lin-Yau (Commun. Math. Phys. 2020), Sun-Wang (Adv. Math. 2022) and Li-Sun-Yang (Calc. Var. Partial Differential Equations 2024), we investigate the existence of Kazdan-Warner type equations on a finite connected graph, based on the theory of Brouwer degree. Specifically, we consider the equation \begin{equation*} -\Delta u=h(x)f(u)-c, \end{equation*} where $h$ is a real-valued function defined on the vertex set $V$, $c\in\mathbb{R}$ and \begin{equation*} f(u)= \left(1-\displaystyle\frac{1}{1+u^{2n}}\right)e^u \end{equation*} with $n\in \mathbb{N}^*$. Different from the previous studies, the main difficulty in this paper is to show that the corresponding equation has only three constant solutions, based on delicate analysis and the connectivity of graphs, which have not been extensively explored in previous literature.

math.AP↗