Yihong Du
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Yihong Du has a total of 146 co-authors. Here are the top co-authors by number of shared publications:
- Wenjie Ni
School of Science and Technology, University of New England, Armidale, Australia
14 shared publications - Edward Norman Dancer
The University of Sydney
13 shared publications Rui Peng
8 shared publications- Bendong Lou
Mathematics and Science College, Shanghai Normal University, Shanghai 200234, China
7 shared publications Zongming Guo
Department of Mathematics, Henan Normal University, Xinxiang, People’s Republic of China
7 shared publicationsWenjie Ni
6 shared publications
Latest Publications
- Open Access
- 1 January 2027
- Published by American Institute of Mathematical Sciences (AIMS) in Discrete and Continuous Dynamical Systems
- Vol. 59, 226-248
Entire positive radial solutions to a class of $ k $-Hessian equations with weight
We consider entire positive radial solutions of $ k $-Hessian equations of the form $ S_k(D^2u) = p(|x|)u^{\alpha}, \ \ x\in \mathbb{R}^n, $ with the weight function $ p(|x|) $ being positive and continuous. For the case $ 0<\alpha\leq k $, we obtain new results on the asymptotic behaviour of every positive radial solution $ u(|x|) $ as $ |x|\to\infty $. When $ p(|x|) $ behaves like $ |x|^{\delta}, \ e^{m|x|}, $ or $ (\ln |x|)^{\theta} $ at infinity, where $ \delta, \ m $, and $ \theta $ are constants in $ \mathbb R $, we reveal how the behaviour of $ u(|x|) $ changes as the parameters vary and, in many cases, we obtain the precise rates of $ u(|x|) $ as $ |x|\to\infty $. Our analysis relies on the construction of suitable upper and lower radial solutions.- 1 January 2027
- Published by American Institute of Mathematical Sciences (AIMS) in Discrete and Continuous Dynamical Systems
- Vol. 59, 226-248
Open AccessEntire positive radial solutions to a class of $ k $-Hessian equations with weight
We consider entire positive radial solutions of $ k $-Hessian equations of the form $ S_k(D^2u) = p(|x|)u^{\alpha}, \ \ x\in \mathbb{R}^n, $ with the weight function $ p(|x|) $ being positive and continuous. For the case $ 0<\alpha\leq k $, we obtain new results on the asymptotic behaviour of every positive radial solution $ u(|x|) $ as $ |x|\to\infty $. When $ p(|x|) $ behaves like $ |x|^{\delta}, \ e^{m|x|}, $ or $ (\ln |x|)^{\theta} $ at infinity, where $ \delta, \ m $, and $ \theta $ are constants in $ \mathbb R $, we reveal how the behaviour of $ u(|x|) $ changes as the parameters vary and, in many cases, we obtain the precise rates of $ u(|x|) $ as $ |x|\to\infty $. Our analysis relies on the construction of suitable upper and lower radial solutions. - Open Access
Precise propagation profile for some monostable free boundary problems in time-periodic media
We consider reaction-diffusion equations of the form \begin{equation*} u_t - d u_{xx} = f(t,u), \quad t>0,\ \ x \in [g(t), h(t)], \end{equation*} where $f(t,u)$ is periodic in $t$ and monostable in $u$, and the interval $[g(t), h(t)]$ represents the one dimensional population range of a species with density $u(t,x)$ at time $t$ and spatial location $x$. The free boundaries $x=g(t)$ and $x=h(t)$ evolve subject to a ``preferred population density" condition at the habitat edges. Analogous to the traveling wave solutions in the corresponding Cauchy problem, semi-wave solutions play a fundamental role in understanding the propagation phenomena governed by the free boundary problem here. But in contrast to the Cauchy problem, where the KPP condition plays a subtle role in the precise approximation of its solution (with compactly supported initial function) by the traveling wave solution with minimal speed, here we prove the existence and uniqueness of a semi-wave in a general monostable setting, and obtain a precise description of the convergence of the solution toward the semi-wave as time goes to infinity, where the KPP condition plays no special role. Previously, such a sharp result was proved for a free boundary model only when $f$ is autonomous ($f=f(u)$, see \cite{D} or \cite{DL15} for a related free boundary model), or a less precise result was obtained in the time-periodic case under an extra strong KPP condition on $f$ (see \cite{MDW}, or \cite{DGP} for a related free boundary model). This work appears to be the first to prove the sharp convergence result for a general monostable free boundary problem in a heterogeneous environment, and we believe the methods developed here should have applications to related free boundary problems in heterogeneous media with nonlinearities more general than those of KPP type.Open AccessPrecise propagation profile for some monostable free boundary problems in time-periodic media
We consider reaction-diffusion equations of the form \begin{equation*} u_t - d u_{xx} = f(t,u), \quad t>0,\ \ x \in [g(t), h(t)], \end{equation*} where $f(t,u)$ is periodic in $t$ and monostable in $u$, and the interval $[g(t), h(t)]$ represents the one dimensional population range of a species with density $u(t,x)$ at time $t$ and spatial location $x$. The free boundaries $x=g(t)$ and $x=h(t)$ evolve subject to a ``preferred population density" condition at the habitat edges. Analogous to the traveling wave solutions in the corresponding Cauchy problem, semi-wave solutions play a fundamental role in understanding the propagation phenomena governed by the free boundary problem here. But in contrast to the Cauchy problem, where the KPP condition plays a subtle role in the precise approximation of its solution (with compactly supported initial function) by the traveling wave solution with minimal speed, here we prove the existence and uniqueness of a semi-wave in a general monostable setting, and obtain a precise description of the convergence of the solution toward the semi-wave as time goes to infinity, where the KPP condition plays no special role. Previously, such a sharp result was proved for a free boundary model only when $f$ is autonomous ($f=f(u)$, see \cite{D} or \cite{DL15} for a related free boundary model), or a less precise result was obtained in the time-periodic case under an extra strong KPP condition on $f$ (see \cite{MDW}, or \cite{DGP} for a related free boundary model). This work appears to be the first to prove the sharp convergence result for a general monostable free boundary problem in a heterogeneous environment, and we believe the methods developed here should have applications to related free boundary problems in heterogeneous media with nonlinearities more general than those of KPP type. - Open Access
- 23 January 2026
- Published by Elsevier in Mathematical Biosciences
- Vol. 394, 109624
Propagation through a barrier: Numerical analysis of a reaction-diffusion model with free boundary
- 23 January 2026
- Published by Elsevier in Mathematical Biosciences
- Vol. 394, 109624
Open AccessPropagation through a barrier: Numerical analysis of a reaction-diffusion model with free boundary
- 23 December 2025
- Published by Elsevier in Journal of Differential Equations
- Vol. 458
A predator-prey model with prey-taxis and free boundary: Well-posedness and long-time dynamics
- 23 December 2025
- Published by Elsevier in Journal of Differential Equations
- Vol. 458
A predator-prey model with prey-taxis and free boundary: Well-posedness and long-time dynamics
- Open Access
- 28 October 2025
- Published by Springer Nature in Journal of Mathematical Biology
- Vol. 91 (6) , 1-52
Invasion dynamics of super invaders: elimination of Allee effects by a strategy at the range boundary
We consider a reaction–diffusion model with free boundaries in one space dimension for a single population species with density u ( t , x ) and population range [ g ( t ), h ( t )]. The equations governing the evolution of the range boundary are deduced from the biological assumption that the species maintains its population density at a fixed positive level at the range boundary by advancing or retreating the fronts. Our mathematical results suggest that the Allee effects are eliminated if the species maintains its population density at suitable levels at the range boundary, namely with such a strategy at the range edge the species can invade the environment successfully with all admissible initial populations, exhibiting the dynamics of super invaders. Numerical simulations are used to help understand what happens if the population density levels at the range boundary are maintained in other ranges.
- 28 October 2025
- Published by Springer Nature in Journal of Mathematical Biology
- Vol. 91 (6) , 1-52
Open AccessInvasion dynamics of super invaders: elimination of Allee effects by a strategy at the range boundary
We consider a reaction–diffusion model with free boundaries in one space dimension for a single population species with density u ( t , x ) and population range [ g ( t ), h ( t )]. The equations governing the evolution of the range boundary are deduced from the biological assumption that the species maintains its population density at a fixed positive level at the range boundary by advancing or retreating the fronts. Our mathematical results suggest that the Allee effects are eliminated if the species maintains its population density at suitable levels at the range boundary, namely with such a strategy at the range edge the species can invade the environment successfully with all admissible initial populations, exhibiting the dynamics of super invaders. Numerical simulations are used to help understand what happens if the population density levels at the range boundary are maintained in other ranges.
- Open Access
Convergence to a receding wave in a monostable free boundary problem
We study a monostable reaction-diffusion equation of the form $u_t=du_{xx}+f(u)$ over a semi-infinite spatial domain $[g(t),\infty)$, with $x=g(t)$ the free boundary whose evolution is governed by equations derived from a ``preferred population density'' principle, which postulates that the species with population density $u(t,x)$ and population range $[g(t),\infty)$ maintains a certain density $δ$ at the habitat edge $x=g(t)$. In the ``high-density'' regime, where $δ$ exceeds the carrying capacity of the favourable environment represented by a monostable function $f(u)$, it is known (see \cite{DLNS} for the case of a bounded population range $[g(t), h(t)]$) that for large time, the front retreats as time advances. In this work, the unboundedness of the population range $[g(t),\infty)$ allows us to prove that, as time $t$ converges to infinity, the free boundary $x=g(t)$ converges to $\infty$ with a constant asymptotic speed $c(δ)>0$ determined by an associated semi-wave problem, and the population density $u(t,x)$ has the property that $u(t,x+g(t))$ converges uniformly to $q_{c(δ)}(x)$, the semi-wave profile function associated with the speed $c(δ)$. It turns out that in the retreating situation considered here, some key techniques developed for advancing fronts in related free boundary models do not work anymore. This difficulty is overcome here by a ``touching method", which uses a family of lower and upper solutions constructed from semi-waves of some carefully designed auxiliary problems to touch the solution $u(t,x)$ at the moving boundary $x=g(t)$, thereby generating a setting where the comparison principle can be used to obtain the desired estimates for $g'(t)$ and $u(t,x)$. We believe this method will find applications elsewhere.Open AccessConvergence to a receding wave in a monostable free boundary problem
We study a monostable reaction-diffusion equation of the form $u_t=du_{xx}+f(u)$ over a semi-infinite spatial domain $[g(t),\infty)$, with $x=g(t)$ the free boundary whose evolution is governed by equations derived from a ``preferred population density'' principle, which postulates that the species with population density $u(t,x)$ and population range $[g(t),\infty)$ maintains a certain density $δ$ at the habitat edge $x=g(t)$. In the ``high-density'' regime, where $δ$ exceeds the carrying capacity of the favourable environment represented by a monostable function $f(u)$, it is known (see \cite{DLNS} for the case of a bounded population range $[g(t), h(t)]$) that for large time, the front retreats as time advances. In this work, the unboundedness of the population range $[g(t),\infty)$ allows us to prove that, as time $t$ converges to infinity, the free boundary $x=g(t)$ converges to $\infty$ with a constant asymptotic speed $c(δ)>0$ determined by an associated semi-wave problem, and the population density $u(t,x)$ has the property that $u(t,x+g(t))$ converges uniformly to $q_{c(δ)}(x)$, the semi-wave profile function associated with the speed $c(δ)$. It turns out that in the retreating situation considered here, some key techniques developed for advancing fronts in related free boundary models do not work anymore. This difficulty is overcome here by a ``touching method", which uses a family of lower and upper solutions constructed from semi-waves of some carefully designed auxiliary problems to touch the solution $u(t,x)$ at the moving boundary $x=g(t)$, thereby generating a setting where the comparison principle can be used to obtain the desired estimates for $g'(t)$ and $u(t,x)$. We believe this method will find applications elsewhere. - Open Access
Lecture notes: Biological propagation via reaction-diffusion equations with nonlocal diffusion and free boundary
These notes are based on the lectures given in a mini-course at VIASM (Vietnam Institute for Advanced Study in Mathematics) 2025 Summer School. They give a brief account of the theory (with detailed proofs) for propagation governed by a nonlocal reaction-diffusion model with free boundaries in one space dimension. The main part is concerned with a KPP reaction term, though the basic results on the existence and uniqueness of solutions as well as on the comparison principles are for more general situations. The contents are mostly taken from published recent works of the author with several collaborators, where the kernel function was assumed to be symmetric: J(x)=J(-x). When J(x) is not symmetric, significant differences may arise in the dynamics of the model, as shown in several preprints quoted in the references at the end of these notes, but many of the existing techniques can be easily extended to cover the "weakly non-symmetric case", and this is done here with all the necessary details.
Open AccessLecture notes: Biological propagation via reaction-diffusion equations with nonlocal diffusion and free boundary
These notes are based on the lectures given in a mini-course at VIASM (Vietnam Institute for Advanced Study in Mathematics) 2025 Summer School. They give a brief account of the theory (with detailed proofs) for propagation governed by a nonlocal reaction-diffusion model with free boundaries in one space dimension. The main part is concerned with a KPP reaction term, though the basic results on the existence and uniqueness of solutions as well as on the comparison principles are for more general situations. The contents are mostly taken from published recent works of the author with several collaborators, where the kernel function was assumed to be symmetric: J(x)=J(-x). When J(x) is not symmetric, significant differences may arise in the dynamics of the model, as shown in several preprints quoted in the references at the end of these notes, but many of the existing techniques can be easily extended to cover the "weakly non-symmetric case", and this is done here with all the necessary details.
- 8 August 2025
- Published by World Scientific Pub Co Pte Ltd in Communications in Contemporary Mathematics
- Vol. 28 (08)
Propagation in a time-periodic environment governed by a reaction–diffusion model of KPP type with a new free boundary condition
In this paper, we investigate the spreading behavior of an invasive species modeled by a KPP type reaction–diffusion equation, in time-periodic medium of one space dimension, with a new free boundary condition that arises from a “preferred population density” assumption at the range boundary. For the model in homogeneous medium, Du [Propagation dynamics of the monostable reaction-diffusion equation with a new free boundary condition, Discrete Contin. Dyn. Syst. 44 (2024) 2524–2563] obtained a fairly complete description of the long-time dynamics and showed that the concerned species always spreads successfully, which contrasts sharply to the corresponding free boundary models in Du and Lin [Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal. 42 (2010) 377–405] and other related works, where vanishing of the species is possible. In this paper, we are interested in the effect of time-periodic perturbation of the environment on the propagation dynamics of such a new free boundary model. Our results show that the main features of the model are retained under the perturbation, namely the species always spreads successfully and there is an asymptotic spreading speed. Furthermore, as the preferred population density at the range boundary approaches zero, we show that the solution of our free boundary problem converges to the solution of the corresponding Cauchy problem.
- 8 August 2025
- Published by World Scientific Pub Co Pte Ltd in Communications in Contemporary Mathematics
- Vol. 28 (08)
Propagation in a time-periodic environment governed by a reaction–diffusion model of KPP type with a new free boundary condition
In this paper, we investigate the spreading behavior of an invasive species modeled by a KPP type reaction–diffusion equation, in time-periodic medium of one space dimension, with a new free boundary condition that arises from a “preferred population density” assumption at the range boundary. For the model in homogeneous medium, Du [Propagation dynamics of the monostable reaction-diffusion equation with a new free boundary condition, Discrete Contin. Dyn. Syst. 44 (2024) 2524–2563] obtained a fairly complete description of the long-time dynamics and showed that the concerned species always spreads successfully, which contrasts sharply to the corresponding free boundary models in Du and Lin [Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal. 42 (2010) 377–405] and other related works, where vanishing of the species is possible. In this paper, we are interested in the effect of time-periodic perturbation of the environment on the propagation dynamics of such a new free boundary model. Our results show that the main features of the model are retained under the perturbation, namely the species always spreads successfully and there is an asymptotic spreading speed. Furthermore, as the preferred population density at the range boundary approaches zero, we show that the solution of our free boundary problem converges to the solution of the corresponding Cauchy problem.
- 4 July 2025
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Applied Mathematics
- Vol. 85 (4) , 1315-1343
Long-Time Dynamics of a Competition Model with Nonlocal Diffusion and Free Boundaries: Chances of Successful Invasion
We aim to understand the dynamics of an invading species via a two species competition model of Lotka–Volterra type with nonlocal diffusions, where the invader with density [Formula: see text] invades the territory (represented by the real line [Formula: see text]) of a native competitor with density [Formula: see text], via two fronts, [Formula: see text] on the left and [Formula: see text] on the right. So the population range of the invader [Formula: see text] is the evolving interval [Formula: see text] and the reaction-diffusion equation for [Formula: see text] has two free boundaries, with [Formula: see text] decreasing in [Formula: see text] and [Formula: see text] increasing in [Formula: see text]. Let [Formula: see text] and [Formula: see text]. In our earlier work [Y. Du, W. Ni, and L. Shi, SIAM J. Math. Anal., 57 (2025), pp. 1195–1226], some detailed descriptions of the long-time dynamics of the model were obtained according to whether [Formula: see text] is [Formula: see text] or finite. In the latter case, we proved that the invader [Formula: see text] vanishes in the long run and [Formula: see text] survives the invasion, while in the former case, we obtained a rather satisfactory description of the long-time asymptotic limits of [Formula: see text] and [Formula: see text] when the parameter [Formula: see text] in the model is less than 1 (meaning that the competition from [Formula: see text] felt by [Formula: see text] is relatively weak). In the current paper, first we obtain sharp criteria to distinguish the case [Formula: see text] from the case [Formula: see text]. Second, for the remaining case [Formula: see text], we reveal complicated but interesting dynamics. For example, when [Formula: see text] and [Formula: see text] is a weak competitor, (a) we obtain biologically meaningful conditions that guarantee the vanishing of the invader [Formula: see text], and (b) we reveal chances for [Formula: see text] to invade successfully; in particular, we demonstrate that both [Formula: see text] and [Formula: see text] are possible (the latter seems to be the first example for this kind of population model, with either local or nonlocal diffusion).
- 4 July 2025
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Applied Mathematics
- Vol. 85 (4) , 1315-1343
Long-Time Dynamics of a Competition Model with Nonlocal Diffusion and Free Boundaries: Chances of Successful Invasion
We aim to understand the dynamics of an invading species via a two species competition model of Lotka–Volterra type with nonlocal diffusions, where the invader with density [Formula: see text] invades the territory (represented by the real line [Formula: see text]) of a native competitor with density [Formula: see text], via two fronts, [Formula: see text] on the left and [Formula: see text] on the right. So the population range of the invader [Formula: see text] is the evolving interval [Formula: see text] and the reaction-diffusion equation for [Formula: see text] has two free boundaries, with [Formula: see text] decreasing in [Formula: see text] and [Formula: see text] increasing in [Formula: see text]. Let [Formula: see text] and [Formula: see text]. In our earlier work [Y. Du, W. Ni, and L. Shi, SIAM J. Math. Anal., 57 (2025), pp. 1195–1226], some detailed descriptions of the long-time dynamics of the model were obtained according to whether [Formula: see text] is [Formula: see text] or finite. In the latter case, we proved that the invader [Formula: see text] vanishes in the long run and [Formula: see text] survives the invasion, while in the former case, we obtained a rather satisfactory description of the long-time asymptotic limits of [Formula: see text] and [Formula: see text] when the parameter [Formula: see text] in the model is less than 1 (meaning that the competition from [Formula: see text] felt by [Formula: see text] is relatively weak). In the current paper, first we obtain sharp criteria to distinguish the case [Formula: see text] from the case [Formula: see text]. Second, for the remaining case [Formula: see text], we reveal complicated but interesting dynamics. For example, when [Formula: see text] and [Formula: see text] is a weak competitor, (a) we obtain biologically meaningful conditions that guarantee the vanishing of the invader [Formula: see text], and (b) we reveal chances for [Formula: see text] to invade successfully; in particular, we demonstrate that both [Formula: see text] and [Formula: see text] are possible (the latter seems to be the first example for this kind of population model, with either local or nonlocal diffusion).