arXiv subjects

Junfan Lu

Publications and source records attributed to Junfan Lu.

4 recordsLinked to original sources

arXiv2026

Regularity of Solutions to One-Dimensional Degenerate Diffusion Equations with Reactions

We study the one-dimensional reaction-diffusion equation \[ u_t=[A(u)]_{xx}+f(x,u). \] The diffusion operator belongs to a broad class of nonlinear degenerate diffusion operators that includes the porous medium operator as a special case. We develop a systematic regularity theory for the solutions and their free boundaries. First, we establish the $C^1$ regularity of the pressure variable $v$, together with a lower bound for its second spatial derivative. Next, we prove Darcy's law and that, after the waiting time, a right (resp.\ left) free boundary moves with strictly positive (resp.\ negative) velocity (Theorem 3.1), thereby strengthening the previously known results which only gave nonnegativity (resp.\ nonpositivity). Finally, under additional structural assumptions on the diffusion and reaction terms, we obtain higher regularity for both the solution and its free boundaries (Theorems 4.5 and 4.6). These results extend several classical regularity properties of the porous medium equation to a much broader class of degenerate diffusion equations with reactions.

math.AP↗

arXiv2025

Fife-McLeod's Theorem for Spatially Periodic Degenerate Diffusion Equations

For one dimensional homogeneous bistable diffusion equations, Fife-McLeod ([Arch. Ration. Mech. Anal., 65 (1977), 335-361]) gave a well-known theorem which says that spreading solutions starting from compactly supported initial data can be exponentially approximated by traveling wave solutions. We will extend this theorem to {\it degenerate diffusion equations in periodic environments}. First, we construct a {\it periodic traveling sharp wave} to the equation, which has a positive profile on the left half-line and a right free boundary governed by the Darcy's law. To achieve this we use a renormalization approach in which crucial uniform gradient estimates near the free boundary are derived via delicate asymptotic analysis. Next we show that the central part of any spreading solution decays exponentially to a periodic steady state. Based on these results, we can construct super- and sub-solutions to prove the Fife-McLeod's theorem for our equation: any spreading solution with compactly supported initial data can be exponentially approximated by the periodic traveling sharp wave.

math.AP↗

arXiv2023

The effect of an unfavorable region on the invasion process of a species

To model a propagating phenomena through the environment with an unfavorable region, we consider a reaction diffusion equation with negative growth rate in the unfavorable region and bistable reaction outside of it. We study rigorously the influence of $L$, the width of the unfavorable region, on the propagation of solutions. It turns out that there exists a critical value $L^*$ depending only on the reaction term such that, when $L L^*$ we have a trichotomy result: spreading/residue happens for a species with large/small initial population, but, for a species with medium-sized initial data, it can not pass through the region either and converges to a transition steady state.

math.AP↗

arXiv2018

Entire Solutions of the Fisher-KPP Equation on the Half Line

In this paper we study the entire solutions of the Fisher-KPP equation $u_t=u_{xx}+f(u)$ on the half line $[0,\infty)$ with Dirichlet boundary condition at $x=0$. (1). For any $c\geq 2\sqrt{f'(0)}$, we show the existence of an entire solution $\mathcal{U}^c(x,t)$ which connects the traveling wave solution $ϕ^c(x+ct)$ at $t=-\infty$ and the unique positive stationary solution $V(x)$ at $t=+\infty$; (2). We also construct an entire solution $\mathcal{U}(x,t)$ which connects the solution of $η_t =f(η)$ at $t=-\infty$ and $V(x)$ at $t=+\infty$.

math.AP↗