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Yuanyang Hu

Publications and source records attributed to Yuanyang Hu.

13 recordsLinked to original sources
arXiv2026

Bramson correction and convergence to the critical wave for Fisher-KPP equations on $\mathbb Z^d$

We consider Fisher-KPP equations with nearest-neighbor diffusion on $\mathbb Z^d$, $d\geq2$, with nonzero finitely supported initial data. We prove a logarithmic delay of the front along each signed coordinate axis and show that the transition region has uniformly bounded width. On every fixed-width half-tube around an axis, the solution converges to translates of the minimal-speed lattice traveling wave, with a bounded phase. We also obtain an upper bound with a logarithmic correction in every direction. The proof uses weighted estimates, bounds for tilted random walks, and a product lower solution. Concavity of the reaction is not assumed.

math.AP↗
arXiv2026

Sharp Propagation and the Local-to-Nonlocal Transition for Fractional Fisher-KPP Equations on Integer Lattices

Let $P$ denote normalized nearest-neighbor averaging on $\mathbb{Z}^d$ and set $A=I-P$. We consider $\partial_tu+A^su=f(u),t>0, x\in\mathbb{Z}^d, 0 0$, compactly supported data have logarithmic propagation rate $a/(d+2s)$. In the logistic case, each fixed level set lies between two constant multiples of $\exp(at/(d+2s))$; this localization remains valid for data with the critical algebraic tail. Nondecreasing front-like data in one dimension instead propagate at rate $a/(2s)$, and the resulting acceleration rules out planar traveling fronts of finite speed in every dimension. We also analyze the singular limit $s\uparrow1$. If $\tau_s=-\log(1-s)$, then throughout $at<\tau_s$ compactly supported solutions follow the Wulff shape of the local nearest-neighbor equation. After a constant multiple of $\tau_s$, their propagation is exponential. Half-space data exhibit the corresponding transition from the directional local speed to the exponent $a/(2s)$. The analysis rests on sharp bounds for the subordinated kernel, algebraic barriers, a compactness--Liouville argument, and a decomposition of the fractional walk into its nearest-neighbor and rare long-jump parts.

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arXiv2025

$L^{p}$-$L^{q}$ estimates of the heat kernels on graphs with applications to a parabolic system

Let $G=(V, E)$ be a locally finite connected graph satisfying curvature-dimension conditions ($CDE(n, 0)$ or its strengthened version $CDE'(n, 0))$) and polynomial volume growth conditions of degree $m$. We systematically establish sharp $L^{p}$-bounds and decay-type $L^{p}$-$L^{q}$ estimates for heat operators on $G$, accommodating both bounded and unbounded Laplacians. The analysis utilizes Li-Yau-type Harnack inequalities and geometric completeness arguments to handle degenerate cases. As a key application, we prove the existence of global solutions to a semilinear parabolic system on $G$ under critical exponents governed by volume growth dimension $m$.

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arXiv2024

Life span of solutions to a semilinear parabolic equation on locally finite graphs

Let $G=(V,E)$ be a locally finite connected graph. We develop the first eigenvalue method on $G$ introduced in 1963 by Kaplan \cite{Kaplan} on Euclidean space, the discrete Phragm\'{e}n-Lindel\"{o}f principle of parabolic equations and upper and lower solutions method on $G$. Using these methods, we establish the estimates and asymptotic behaviour of the life span of solutions to a semilinear heat equation with initial data $\lambda\psi(x)$ for different scales of $\lambda$ on $G$ under some different conditions. Our results are different from the continuous case, which is related to the structure of the graph $G$.

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arXiv2023

Existence and uniqueness of solutions to Bogomol'nyi equations on graphs

Let $G=(V,E)$ be a connected finite graph. We study the Bogomol'nyi equation \begin{equation*} Δu= \mathrm{e}^{u}-1 +4 π\sum_{s=1}^{k} n_s δ_{z_{s}} \quad \text { on } \quad G, \end{equation*} where $z_1, z_2,\dots, z_k$ are arbitrarily chosen distinct vertices on the graph, $n_j$ is a positive integer, $j=1,2,\cdots, k$ and $δ_{z_{s}}$ is the Dirac mass at $z_s$. We obtain a necessary and sufficient condition for the existence and uniqueness of solutions to the Bogomol'nyi equation.

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arXiv2022

Prey-Predator models on graphs

In this paper, we study the Lotka-Volterra prey-predator models consisting of two species on finite connected graphs under Neumann condition and the condition that there is no boundary condition. We establish the global stability of the unique constant equilibrium solution of each parabolic system.

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arXiv2022

Lotka-Volterra competition models on finite graphs

In this paper, we study three two competing species Lotka-Volterra competition models on finite connected graphs, with Dirichlet, Neumann or no boundary conditions. We get that when time goes to infinity, either one specie extincts while the other becomes surviving or both competing species coexist, which depend crucially on the strengh of species' competitiveness and the size of the initial population under the Neumann boundary condition and the condition that there is no boundary condition. One of our results partially answer a question posed by Slav\'ık in [SIAM J. Appl. Dyn. Syst., 19 (2020)]. The critical techiniques in the proof of our main results are upper and lower solutions method, which are developed for weakly coupled parabolic systems on finite graphs in this article.

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arXiv2022

Monotone methods for semilinear parabolic and elliptic equations on graphs

This paper is devoted to investigate the extinction and propagation properties of solutions to the graph Laplacian parabolic problems with Kpp type or Allen-Cahn type forcing terms on graphs. To this end, we establish the (strong) maximum principle and the upper and lower solutions method for parabolic and elliptic problems on graphs. The stability of equilibrium solutions is studied by constructing suitable upper and lower solutions. Moreover, we give an example and numerical experiments to demonstrate one of our main results.

math.AP↗
arXiv2022

Existence of solutions to a generalized self-dual Chern-Simons equation on finite graphs

Let $G=(V,E)$ be a connected finite graph. We study the existence of solutions for the following generalized Chern-Simons equation on $G$ \begin{equation*} Δu=λ\mathrm{e}^{u}\left(\mathrm{e}^{u}-1\right)^{5}+4 π\sum_{s=1}^{N} δ_{p_{s}} \quad , \end{equation*} where $λ>0$, $δ_{p_{s}}$ is the Dirac mass at the vetex $p_s$, and $p_1, p_2,\dots, p_N$ are arbitrarily chosen distinct vertices on the graph. We show that there exists a critial value $\hatλ$ such that when $λ> \hatλ$, the generalized Chern-Simons equation has at least two solutions, when $λ= \hatλ$, the generalized Chern-Simons equation has a solution, and when $λ< \hatλ$, the generalized Chern-Simons equation has no solution.

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arXiv2019

A free boundary problem for spreading under shifting climate

In this paper we consider a free boundary problem which models the spreading of an invasive species whose spreading is enhanced by the changing climate. We assume that the climate is shifting with speed c and obtain a complete classification of the long-time dynamical behaviour of the species. The model is similar to that in [9] with a slight refinement in the free boundary condition. While [9], like many works in the literature, investigates the case that unfavourable environment is shifting into the favourable habitat of the concerned species, here we examine the situation that the unfavourable habitat of an invasive species is replaced by a favourable environment with a shifting speed c. We show that a spreading-vanishing dichotomy holds, and there exists a critical speed$c_0$ such that when spreading happens in the case $c < c_0$, the spreading profile is determined by a semi-wave with forced speed c, but when $c \geq c_0$, the spreading profile is determined by the usual semi-wave with speed $c_0$.

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