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Maolin Zhou

Publications and source records attributed to Maolin Zhou.

At least 19 recordsLinked to original sources
arXiv2026

Asymmetric Periodic Water Waves in Finite Depth

We consider the full two-dimensional water-wave equations in finite depth, with and without surface tension. Water waves have been studied mathematically for more than two centuries. Reflection symmetry remains a common assumption in constructions of periodic steady waves, although asymmetric profiles have been observed in experiments and numerical computations. We construct families of periodic stationary solutions whose free surfaces have no axis of reflection, in both the pure-gravity and capillary--gravity cases. To the best of our knowledge, this provides the first rigorous construction of asymmetric periodic pure-gravity waves for the full two-dimensional water-wave problem in finite depth. The proof combines localized vorticity laws with a Lyapunov--Schmidt reduction. In the pure-gravity case, we introduce a small background shear flow to obtain an invertible boundary operator and couple the interior and free-surface approximations at leading order.

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arXiv2026

Propagation Direction of Bistable Traveling Fronts in the Lotka--Volterra Competition--Diffusion System

We study the propagation direction of bistable traveling fronts in the two-species Lotka--Volterra competition--diffusion system under strong competition. A complete characterization of the sign of the wave speed has remained a long-standing unsolved problem. We establish the first global necessary and sufficient criterion for zero wave speed by combining a Maxwell-type identity with a phase-plane rigidity argument. This criterion yields a unique zero-speed threshold, identifies its value in the symmetric case and its limiting values in two extreme regimes, and consequently determines the propagation direction throughout the strong-competition parameter region.

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arXiv2026

Sharp logarithmic corrections for a strong-weak Lotka-Volterra competition system

We study the one-dimensional strong-weak Lotka-Volterra competition-diffusion system \[ u_t=u_{xx}+u(1-u-av),\qquad v_t=dv_{xx}+rv(1-v-bu), \] with compactly supported initial data under the parameter condition $0<a<1<b$. Existing literature only gives leading-order spreading speed asymptotics without refined logarithmic corrections for wave fronts over the full parameter space. We convert the competitive system into an equivalent cooperative parabolic system and establish moving-domain comparison principles. Based on heat-kernel estimates, we derive sharp logarithmic asymptotic expansions for the rightmost level set of the stronger species $u$. According to the magnitudes of three characteristic speeds, five long-time dynamical regimes are classified, including pulled, nonlocally pulled, pushed and critical transition fronts, with explicit logarithmic and double-logarithmic phase corrections. Our results capture delicate long-time phase offsets of wave profiles neglected in previous leading-order theories.

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arXiv2026

Sharp Thresholds for the Porous Medium Equation with a Combustion Reaction in Higher Dimensions

We study the porous medium equation with a combustion-type reaction, \[ u_t=Δu^m+f(u),\qquad x\in\mathbb R^N,\ t>0, \] for radial, nonnegative, compactly supported initial data. A complete classification of the long-time behaviour of bounded solutions is established. In dimension $N=2$, every such solution converges locally uniformly to one of the constants $0$, $θ$, or $1$; for ordered families of initial data there exists a unique threshold parameter separating vanishing from spreading, and the critical solution converges to the ignition temperature $θ$. In dimensions $N\ge3$, under a natural total-disconnectedness condition on the set of central values of ground states, every bounded solution converges locally uniformly to either $0$, $θ$, $1$, or a radial ground state $U\in\mathcal S$. Moreover, the ignition state $θ$ is excluded as a transition limit in $N\ge3$ via a novel normalized annular perturbation argument. For transition solutions, we provide estimates for the propagation speed of the free boundary: in dimension two, \[ b(t)\asymp \frac{\sqrt t}{(\log t)^{\frac{m-1}{2m}}}, \] and in dimensions $N\ge3$, whenever the limit is a ground state, \[ b(t)\asymp t^{\frac{m}{N(m-1)+2}}. \] These results reveal a sharp dimensional dichotomy in the degenerate combustion dynamics.

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arXiv2025

The principal eigenvalue of an age-structured operator with diffusion and advection: qualitative analysis and an application

In this paper, we investigate an eigenvalue problem associated with an age-structured operator incorporating random diffusion and advection. Our primary focus is on examining the asymptotic behaviors of the principal eigenvalue with respect to large advection and small or large diffusion rates. We subsequently apply these results to a nonlinear age-structured model, providing a better understanding of how diffusion and advection influence the spatial distribution of species. Among other ingredients, our approach involves constructing various types of super- and sub-solutions to tackle the novel challenges posed by the nonlocal terms in the problems under consideration.

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arXiv2025

Non-convergence of the principal eigenvalue of elliptic operators for large advection

This paper investigates the limit of the principal eigenvalue $λ(s)$ as $s\to+\infty$ for the following elliptic equation \begin{align*} -Δφ(x)-2s\mathbf{v}\cdot\nablaφ(x)+c(x)φ(x)=λ(s)φ(x), \quad x\in Ω \end{align*} in a bounded domain $Ω\subset \mathbb{R}^d (d\geq 1)$ with the Neumann boundary condition. Previous studies have shown that under certain conditions on $\mathbf{v}$, $λ(s)$ converges as $s\to\infty$ (including cases where $\lim\limits_{s \to\infty }λ(s)=\pm\infty$). This work constructs an example such that $λ(s)$ is divergent as $s\to+\infty$. This seems to be the first rigorous result demonstrating the non-convergence of the principal eigenvalue for second-order linear elliptic operators with some strong advection. As an application, we demonstrate that for the classical advection-reaction-diffusion model with advective velocity field $\mathbf{v}=\nabla m$, where $m$ is a potential function with infinite oscillations, the principal eigenvalue changes sign infinitely often along a subsequence of $s\to\infty$. This leads to solution behaviors that differ significantly from those observed when $m$ is non-oscillatory.

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arXiv2025

Linear vs. nonlinear speed selection of the front propagation into unstable states

In this paper, we mainly consider the speed selection problem for the classical Lotka-Volterra competition system. For the first time, we propose a sufficient and necessary condition for this long-standing problem from a new point of view. Moreover, our results can also reveal the essence of the linearly selected problem for the monostable dynamical system from the observation of the decay rate of the minimal traveling wave solution.

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arXiv2025

Nonlocal to Local Convergence of Stefan Problems Under Optimal Convergence Condition

In this paper, we consider a free boundary problem with a nonlocal diffusion kernel function $k(x)$. Due to the long distance exchange effect of nonlocal diffusion, the free boundary can expand discontinuously, which makes the problem rather complicated. Among other things, we propose the optimal convergence condition without assuming the symmetry or compactness of $k$, i.e., the Fourier transform of $k$ satisfies $$\hat{k}(ξ)=1-|ξ|^2+o(|ξ|^2)\ \ \mbox{ as }ξ\rightarrow 0,$$ and discover an equivalent characterization of this optimal condition. More importantly, by the employment of the variational inequality, the apriori estimates and the Fourier transform, we demonstrate that, along a series of properly rescaled kernel functions, the corresponding solutions to the nonlocal free boundary problems converge to the solution of the classical Stefan problem under the proposed optimal condition.

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arXiv2024

On the propagation speed of the single monostable equation

In this paper, we first focus on the speed selection problem for the reaction-diffusion equation of the monostable type. By investigating the decay rates of the minimal traveling wave front, we propose a sufficient and necessary condition that reveals the essence of propagation phenomena. Moreover, since our argument relies solely on the comparison principle, it can be extended to more general monostable dynamical systems, such as nonlocal diffusion equations.

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arXiv2024

Exact blow-up profiles for the parabolic-elliptic Keller-Segel system in dimensions $N\ge 3$

In this paper, we obtain the exact blow-up profiles of solutions of the Keller-Segel-Patlak system in the space with dimensions $N\ge 3$, which solves an open problem proposed by P. Souplet and M. Winkler in 2019. To establish this achievement, we develop the zero number argument for nonlinear equations with unbounded coefficients and construct a family of auxiliary backward self-similar solutions through nontrivial ODE analysis.

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arXiv2024

Principal eigenvalue for some elliptic operators with large drift: Neumann boundary conditions

The paper is concerned with the principal eigenvalue of some linear elliptic operators with drift in two dimensional space. We provide a refined description of the asymptotic behavior for the principal eigenvalue as the drift rate approaches infinity. Under some non-degeneracy assumptions, our results illustrate that these asymptotic behaviors are completely determined by some connected components in the omega-limit set of the system of ordinary differential equations associated with the drift term, which includes stable fixed points, stable limit cycles, hyperbolic saddles connecting homoclinic orbits, and families of closed orbits. Some discussions on degenerate cases are also included.

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arXiv2024

Convergence of Solutions of the Porous Medium Equation with Reactions

Consider the Cauchy problem of one dimensional porous medium equation (PME) with reactions. We first prove a general convergence result, that is, any bounded global solution starting at a nonnegative compactly supported initial data converges as $t\to \infty$ to a nonnegative zero of the reaction term or a ground state stationary solution. Based on it, we give out a complete classification on the asymptotic behaviors of the solutions for PME with monostable, bistable and combustion types of nonlinearities.

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arXiv2024

Degenerate bifurcations of two-fold doubly-connected uniformly rotating vortex patches

In this paper, we obtain families of two-fold doubly-connected uniformly rotating vortex patches of the 2-D incompressible Euler equations emanating from some specific annuli. The main difficulty comes from strong degeneracy of the problem, neither the kernel of linearization is one-dimensional nor the transeversallity condition holds. To this end, we make a detailed analysis on the nonlinear functional and the bifurcation curves are obtained by perturbing real algebraic varieties defined by truncated polynomials. In addition, our result partially answers an problem proposed by Hmidi and Mateu in \cite{Hmidi2016a} (\emph{Adv.Math.302 (2016), 799-850}).

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arXiv2023

A Counterexample for the Principal Eigenvalue of An Elliptic Operator with Large Advection

There are numerous studies focusing on the convergence of the principal eigenvalue $λ(s)$ as $s\to+\infty$ corresponding to the elliptic eigenvalue problem \begin{align*} -Δφ(x)-2s\mathbf{v}\cdot\nablaφ(x)+c(x)φ(x)=λ(s)φ(x),\quad x\in Ω, \end{align*} where $Ω$ is a bounded domain and the advection term $\mathbf{v}$ under some certain restrictions. In this paper, we construct an infinitely oscillating gradient advection term $\mathbf{v}=\nabla m(x)\in C^1(Ω)$ such that the principal eigenvalue $λ(s)$ does not converge as $s\to+\infty$. As far as we know, this is the first result that guarantee the non-convergence of the principal eigenvalue.

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arXiv2023

Novel Spatial Profiles of Population Distribution of Two Diffusive SIS Epidemic Models with Mass Action Infection Mechanism and Small Movement Rate for the Infected Individuals

In this paper, we are concerned with two SIS epidemic reaction-diffusion models with mass action infection mechanism of the form $SI$, and study the spatial profile of population distribution as the movement rate of the infected individuals is restricted to be small. For the model with a constant total population number, our results show that the susceptible population always converges to a positive constant which is indeed the minimum of the associated risk function, and the infected population either concentrates at the isolated highest-risk points or aggregates only on the highest-risk intervals once the highest-risk locations contain at least one interval. In sharp contrast, for the model with a varying total population number which is caused by the recruitment of the susceptible individuals and death of the infected individuals, our results reveal that the susceptible population converges to a positive function which is non-constant unless the associated risk function is constant, and the infected population may concentrate only at some isolated highest-risk points, or aggregate at least in a neighborhood of the highest-risk locations or occupy the whole habitat, depending on the behavior of the associated risk function and even its smoothness at the highest-risk locations. Numerical simulations are performed to support and complement our theoretical findings.

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arXiv2022

Optimisation of total population in logistic model with nonlocal dispersals and heterogeneous environments

In this paper, we investigate the issue of maximizing the total equilibrium population with respect to resources distribution m(x) and diffusion rates d under the prescribed total amount of resources in a logistic model with nonlocal dispersals. Among other things, we show that for $d\ge1$, there exist $C_0, C_1>0$, depending on the $\|m\|_{L^1}$ only, such that $$C_0\sqrt{d}<\mbox{supremum~ of~ total~ population}<C_1\sqrt{d}.$$ However, when replaced by random diffusion, a conjecture, proposed by Ni and justified in [3], indicates that in the one-dimensional case, supremum of total population$=3\|m\|_{L^1}$. This reflects serious discrepancies between models with local and nonlocal dispersal strategies.

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