arXiv subjects

Wei-Jie Sheng

Publications and source records attributed to Wei-Jie Sheng.

8 recordsLinked to original sources

arXiv2026

Transition fronts of combustion reaction-diffusion equations in domains with multiple cylindrical branches

This paper is concerned with propagation dynamics for combustion reaction-diffusion equations in domains with multiple cylindrical branches. We first establish the existence and uniqueness of a time-increasing entire solution behaving like planar traveling fronts in some branches and converging to $0$ in the remaining part of the domain as $t\to-\infty$. Under the assumption of complete propagation, we then show that this entire solution propagates into the other branches in the form of planar traveling fronts (up to finite shifts) and converges to $1$ elsewhere as $t\to+\infty$. In particular, it is proved that this entire solution is a transition front connecting $0$ and $1$, whose global mean speed coincides with the planar wave speed. By assuming complete propagation for front-like solutions originating from single branch, we further prove that every transition front connecting $0$ and $1$ propagates completely. Moreover, we show that the global mean speed is independent of the choice of transition front. Namely, all transition fronts connecting $0$ and $1$ share the same global mean speed.Finally, we provide two sufficient geometric conditions under which the complete propagation assumptions are satisfied.

math.AP↗

arXiv2026

Propagation phenomena of spatially periodic combustion reaction-diffusion equations around an obstacle

This paper is concerned with propagation phenomena of spatially periodic combustion reaction-diffusion equations in exterior domains. It is known that there is a pulsating front connecting 0 and 1 with positive speed in $\mathbb{R}^N$ for any direction $e\in\mathbb{S}^{N-1}$. We first prove that there exists an entire solution originating from a pulsating front in the exterior domain. Then, we prove that the entire solution propagates completely. Additionally, by constructing appropriate super- and sub-solutions, we establish that the entire solution is a transition front connecting 0 and 1, and that it is trapped between two translates of the pulsating front as $t\rightarrow +\infty$. Finally, under a suitable assumption, we show that the entire solution converges to the same pulsating front as $t\rightarrow +\infty$, as well as the uniqueness of such entire solutions.

math.AP↗

arXiv2026

The speeds of propagation for the monostable Lotka-Volterra competition-diffusion system in general unbounded domains

This paper is concerned with the speeds of propagation for the monostable Lotka-Volterra competition-diffusion system in general unbounded domains of $\mathbb{R}^N$. We first establish various definitions of spreading speeds at large time in the situation where one species is an invader and the other is a resident. Then, we study fundamental properties of these new definitions, including their relationships and their dependence on the geometry of the domain and the initial values. Under the conditions that both species possess the same diffusion ability and that the interactions between them are sufficiently weak, we derive an upper bound for the spreading speeds in a large class of domains. Furthermore, we obtain general upper and lower bounds for spreading speeds in exterior domains, as well as a general lower bound in domains containing large half-cylinders. Finally, we construct some particular domains for which the spreading speeds can be zero or infinite.

math.AP↗

arXiv2026

Transition fronts of monotone bistable reaction-diffusion systems around an obstacle

This paper is concerned with the interaction between a planar traveling front and a compact obstacle for monotone bistable reaction-diffusion systems in exterior domains. By constructing appropriate sub- and supersolutions, we first establish the existence, uniqueness and monotonicity of the entire solution emanating from a planar traveling front. In particular, we verify that regardless of the shape of the obstacle, the entire solution locally converges to a stationary solution as time tends to infinity. Under the complete propagation assumption, we further show that the entire solution recovers to the same planar traveling front as time tends to infinity after passing the obstacle, and it constitutes a transition front. In addition, we provide some geometric conditions on the obstacle to ensure that the complete propagation assumption is nonempty. Finally, we apply our theoretical results to the Lotka-Volterra competition-diffusion system.

math.AP↗

arXiv2026

V-shaped transition fronts of monotone bistable reaction-diffusion systems in exterior domains

This paper investigates the propagation phenomena of a monotone bistable reaction-diffusion system in an exterior domain of R2. By constructing suitable sub- and supersolutions, we establish the existence and monotonicity of an entire solution originating from a V-shaped traveling front. It is further shown that, under the complete propagation condition, this entire solution eventually recovers its V-shaped profile as time tends to infty after passing the obstacle. In particular, we show that the entire solution is a V-shaped transition front whose global mean speed coincides with the planar wave speed.

math.AP↗

arXiv2026

Curved fronts of combustion reaction-diffusion equations

This paper is concerned with curved fronts of combustion reaction-diffusion equations in $\mathbb{R}^N$ $(N\geq2)$. By mixing finite planar fronts and constructing suitable super- and subsolutions, we prove the existence, uniqueness and stability of polytope-like curved fronts in $\mathbb{R}^N$. Besides, we show that these curved fronts are transition fronts.

math.AP↗

arXiv2025

Existence and stability of curved fronts for spatially periodic combustion reaction-diffusion equations in $\mathbb{R}^N$

This paper is concerned with curved fronts of combustion reaction-diffusion equations in spatially periodic media in $\mathbb{R}^N$ $(N\geq2)$. Under the assumption that there are moving pulsating fronts for any given propagation direction $e \in \mathbb{S}^{N-1}$, and by constructing suitable super- and sub-solutions, we prove the existence of a curved front with polytope-like shape in $\mathbb{R}^N$. Then we show that the curved front is unique and asymptotically stable.

math.AP↗

arXiv2018

On the mean speed of bistable transition fronts in unbounded domains

This paper is concerned with the existence and further properties of propagation speeds of transition fronts for bistable reaction-diffusion equations in exterior domains and in some domains with multiple cylindrical branches. In exterior domains we show that all transition fronts with complete propagation propagate with the same global mean speed, which turns out to be equal to the uniquely defined planar speed. In domains with multiple cylindrical branches, we show that the solutions emanating from some branches and propagating completely are transition fronts propagating with the unique planar speed. We also give some geometrical and scaling conditions on the domain, either exterior or with multiple cylindrical branches, which guarantee that any transition front has a global mean speed.

math.AP↗