arXiv subjects

Bendong Lou

Publications and source records attributed to Bendong Lou.

At least 19 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.

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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.

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arXiv2025

Asymptotic Behavior of Solutions of a Degenerate Diffusion Equation with a Multistable Reaction

We consider a generalized degenerate diffusion equation with a reaction term $u_t=[A(u)]_{xx}+f(u)$, where $A$ is a smooth function satisfying $A(0)=A'(0)=0$ and $A(u),\ A'(u),\ A''(u)>0$ for $u>0$, $f$ is of monostable type in $[0,s_1]$ and of bistable type in $[s_1,1]$. We first give a trichotomy result on the asymptotic behavior of the solutions starting at compactly supported initial data, which says that, as $t\to \infty$, either small-spreading (which means $u$ tends to $s_1$), or transition, or big-spreading (which means $u$ tends to $1$) happens for a solution. Then we construct the classical and sharp traveling waves (a sharp wave means a wave having a free boundary which satisfies the Darcy's law) for the generalized degenerate diffusion equation, and then using them to characterize the spreading solution near its front.

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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

Singular Limits of Porous Media Equations with Bistable Reactions

We consider a porous media equation with balanced bistable reactions, equipped with some general nonlinear boundary condition. When the coefficient of the reaction term is much larger than that of the diffusion term, we see that, besides the possible free boundary, sharp interfaces appear between two stable steady states. By using the method of matched asymptotic expansions, we derive the motion law of each interface, which is a mean curvature flow (may depends on normal direction of the interface). In addition, the original boundary condition reduces to Robin ones at the points where the interface contacts the domain boundary.

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arXiv2024

Traveling Sharp Waves of Porous Media Equations in Spatially Periodic Environment

We consider one dimensional porous media equations in spatially periodic environment. We will construct a periodic traveling sharp wave whose profile tends to a positive steady state at left infinity and takes zero on the right half line, with a free boundary satisfying the Darcy's law. Our method is to take the limit for a sequence of normalized solutions starting at Heaviside type of initial data. The crucial step is to give a uniform positive lower bound for the instantaneous speed of the free boundary.

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arXiv2024

A Mean Curvature Flow with Prescribed Contact Angles in a High Dimensional Cylinder

In this paper we consider a mean curvature flow $V=H+A$ in a high dimensional cylinder $Ω\times \R$, where, $A$ is a constant, $Ω$ is a bounded domain in $\R^n$, and, for a hypersurface $y=u(x,t)$ over $Ω$, $V$ and $H$ denote its normal velocity and mean curvature, respectively. Assume the hypersurface contacts the cylinder boundary $\partial Ω\times \R$ with prescribed angle $θ(x)$. Under certain assumptions such as $Ω$ is strictly convex and $\|\cosθ\|_{C^2}$ is small, or $Ω$ is not necessarily convex but $|A|$ is sufficiently large, we derive some {\it uniform-in-time gradient bounds} for the solutions to initial boundary value problems. Then, we present a trichotomy result as well as its criterion for the asymptotic behavior of the solutions, that is, when $I:= A|Ω|+\int_{\partial Ω} \cosθ(x) dσ>0$ (resp. $=0$, $<0$), the solution $u$ converges as $t\to \infty$ to a translating solution with positive speed (resp. stationary solution, a translating solution with negative speed).

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arXiv2023

Dimension-Dependent Asymptotic Dynamics for Mean Curvature Flow with Robin Boundary Conditions

We consider a graphical mean curvature flow in a cylinder with Robin boundary conditions, which arises as a geometric model for interface motion in the singular limit of the Allen--Cahn equation with nonlinear boundary conditions. It was shown in [26] that, in the planar case, every solution converges to a translating Grim Reaper with a \emph{fixed profile} and \emph{finite speed}. In this paper, we investigate the radially symmetric problem in higher dimensions and reveal a completely different asymptotic dynamics caused by the spatial dimension. In contrast to the planar case, there is no fixed translating profile governing the long-time behaviour. Instead, the solution propagates with an exponentially increasing speed, while both the gradient $|Du|$ (away from the center) and the instantaneous speed $u_t$ diverge exponentially as $t\to\infty$. This reveals a fundamentally different asymptotic behaviour induced by the interaction between the Robin boundary condition and the spatial dimension, that is, the translating profile continuously degenerates and becomes asymptotically ray-like. Since the equation becomes asymptotically degenerate and no uniform-in-time $C^0$, $C^1$, or $C^2$ estimates are available, our analysis relies on a new approach based on the zero number argument.

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arXiv2022

Translating Solutions of a Generalized Mean Curvature Flow in a Cylinder: I. Constant Boundary Angles

We study a generalized mean curvature flow involving a positive power of the mean curvature and a driving force. In this paper, we first construct all kinds of radially symmetric translating solutions, and then select one of them to satisfy a prescribed boundary angle in a cylinder. We then consider the flow starting at an initial hypersurface: showing the a priori estimates (especially the uniform-in-time bounds for the mean curvature which guarantee the uniform parabolicity of the corresponding fully nonlinear equation), giving the global existence for the solution of the initial boundary value problem, and proving its convergence to the corresponding translating solution. Our study provides a complete exposition on the influence of the dimension, the power of the mean curvature, the driving force and the boundary angles on the existence and stability of radially symmetric translating solutions.

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arXiv2020

Convergence to the Grim Reaper for a Curvature Flow with Unbounded Boundary Slopes

We consider a curvature flow $V=H$ in the band domain $Ω:=[-1,1]\times \R$, where, for a graphic curve $Γ_t$, $V$ denotes its normal velocity and $H$ denotes its curvature. If $Γ_t$ contacts the two boundaries $\partial_\pm Ω$ of $Ω$ with constant slopes, in 1993, Altschular and Wu \cite{AW1} proved that $Γ_t$ converges to a {\it grim reaper} contacting $\partial_\pm Ω$ with the same prescribed slopes. In this paper we consider the case where $Γ_t$ contacts $\partial_\pm Ω$ with slopes equaling to $\pm 1$ times of its height. When the curve moves to infinity, the global gradient estimate is impossible due to the unbounded boundary slopes. We first consider a special symmetric curve and derive its uniform interior gradient estimates by using the zero number argument, and then use these estimates to present uniform interior gradient estimates for general non-symmetric curves, which lead to the convergence of the curve in $C^{2,1}_{loc} ((-1,1)\times \R)$ topology to the {\it grim reaper} with span $(-1,1)$.

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arXiv2019

The Fisher-KPP equation over simple graphs: Varied persistence states in river networks

In this article, we study the growth and spread of a new species in a river network with two or three branches via the Fisher-KPP advection-diffusion equation over some simple graphs with every edge a half infinite line. We obtain a rather complete description of the long-time dynamical behavior for every case under consideration, which can be loosely described by a trichotomy (see Remark 1.7), including two different kinds of persistence states as parameters vary. The phenomenon of "persistence below carrying capacity" revealed here appears new, which does not occur in related models of the existing literature where the river network is represented by graphs with finite-lengthed edges, or the river network is simplified to a single infinite line.

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arXiv2019

Propagation of a Mean Curvature Flow in a Cone

We consider a mean curvature flow in a cone, that is, a hypersurface in a cone which moves toward the opening with normal velocity equaling to the mean curvature, and the contact angle between the hypersurface and the cone boundary being $\varepsilon$-periodic in its position. First, by constructing a family of self-similar solutions, we give a priori estimates for the radially symmetric solutions and prove the global existence. Then we consider the homogenization limit as $\ve\to 0$, and use {\it the slowest self-similar solution} to characterize the solution, with error $O(1)\ve^{1/6}$, in some finite time interval.

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arXiv2019

The Zero Number Diminishing Property under General Boundary Conditions

The so-called {\it zero number diminishing property} (or {\it zero number argument}) is a powerful tool in qualitative studies of one dimensional parabolic equations, which says that, under the zero- or non-zero-Dirichlet boundary conditions, the number of zeroes of the solution $u(x,t)$ of a linear equation is finite, non-increasing and strictly decreasing when there are multiple zeroes (cf. \cite{Ang}). In this paper we extend the result to the problems with more general boundary conditions: $u= 0$ sometime and $u\not= 0$ at other times on the domain boundaries. Such results can be applied in particular to parabolic equations with Robin and free boundary conditions.

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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 $t=+\infty$.

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arXiv2016

A diffusive Fisher-KPP equation with free boundaries and time-periodic advections

We consider a reaction-diffusion-advection equation of the form: $u_t=u_{xx}-β(t)u_x+f(t,u)$ for $x\in (g(t),h(t))$, where $β(t)$ is a $T$-periodic function representing the intensity of the advection, $f(t,u)$ is a Fisher-KPP type of nonlinearity, $T$-periodic in $t$, $g(t)$ and $h(t)$ are two free boundaries satisfying Stefan conditions. This equation can be used to describe the population dynamics in time-periodic environment with advection. Its homogeneous version (that is, both $β$ and $f$ are independent of $t$) was recently studied by Gu, Lou and Zhou \cite{GLZ}. In this paper we consider the time-periodic case and study the long time behavior of the solutions. We show that a vanishing-spreading dichotomy result holds when $β$ is small; a vanishing-transition-virtual spreading trichotomy result holds when $β$ is a medium-sized function; all solutions vanish when $β$ is large. Here the partition of $β(t)$ is much more complicated than the case when $β$ is a real number, since it depends not only on the "size" $\barβ:= \frac{1}{T}\int_0^T β(t) dt$ of $β(t)$ but also on its "shape" $\tildeβ(t) := β(t) - \barβ$.

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arXiv2015

Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries

We consider Fisher-KPP equation with advection: $u_t=u_{xx}-βu_x+f(u)$ for $x\in (g(t),h(t))$, where $g(t)$ and $h(t)$ are two free boundaries satisfying Stefan conditions. This equation is used to describe the population dynamics in advective environments. We study the influence of the advection coefficient $-β$ on the long time behavior of the solutions. We find two parameters $c_0$ and $β^*$ with $β^*>c_0>0$ which play key roles in the dynamics, here $c_0$ is the minimal speed of the traveling waves of Fisher-KPP equation. More precisely, by studying a family of the initial data $\{ σϕ\}_{σ>0}$ (where $ϕ$ is some compactly supported positive function), we show that, (1) in case $β\in (0,c_0)$, there exists $σ^*\geqslant0$ such that spreading happens when $σ> σ^*$ and vanishing happens when $σ\in (0,σ^*]$; (2) in case $β\in (c_0,β^*)$, there exists $σ^*>0$ such that virtual spreading happens when $σ>σ^*$ (i.e., $u(t,\cdot;σϕ)\to 0$ locally uniformly in $[g(t),\infty)$ and $u(t,\cdot + ct;σϕ)\to 1$ locally uniformly in $\R$ for some $c>β-c_0$), vanishing happens when $σ\in (0,σ^*)$, and in the transition case $σ=σ^*$, $u(t, \cdot+o(t);σϕ)\to V^*(\cdot-(β-c_0)t )$ uniformly, the latter is a traveling wave with a "big head" near the free boundary $x=(β-c_0)t$ and with an infinite long "tail" on the left; (3) in case $β= c_0$, there exists $σ^*>0$ such that virtual spreading happens when $σ> σ^*$ and $u(t,\cdot;σϕ)\to 0$ uniformly in $[g(t),h(t)]$ when $σ\in (0,σ^*]$; (4) in case $β\geqslant β^*$, vanishing happens for any solution.

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arXiv2015

Nonlinear diffusion problems with free boundaries: Convergence, transition speed and zero number arguments,

This paper continues the investigation of Du and Lou (J. European Math Soc, to appear), where the long-time behavior of positive solutions to a nonlinear diffusion equation of the form $u_t=u_{xx}+f(u)$ for $x$ over a varying interval $(g(t), h(t))$ was examined. Here $x=g(t)$ and $x=h(t)$ are free boundaries evolving according to $g'(t)=-μu_x(t, g(t))$, $h'(t)=-μu_x(t,h(t))$, and $u(t, g(t))=u(t,h(t))=0$. We answer several intriguing questions left open in the paper of Du and Lou.First we prove the conjectured convergence result in the paper of Du and Lou for the general case that $f$ is $C^1$ and $f(0)=0$. Second, for bistable and combustion types of $f$, we determine the asymptotic propagation speed of $h(t)$ and $g(t)$ in the transition case. More presicely, we show that when the transition case happens, for bistable type of $f$ there exists a uniquely determined $c_1>0$ such that $\lim_{t\to\infty} h(t)/\ln t=\lim_{t\to\infty} -g(t)/\ln t=c_1$, and for combustion type of $f$, there exists a uniquely determined $c_2>0$ such that $\lim_{t\to\infty} h(t)/\sqrt t=\lim_{t\to\infty} -g(t)/\sqrt t=c_2$. Our approach is based on the zero number arguments of Matano and Angenent, and on the construction of delicate upper and lower solutions.

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arXiv2014

Long time behavior of solutions of a reaction-diffusion equation on unbounded intervals with Robin boundary conditions

We study the long time behavior, as $t\to\infty$, of solutions of $$ \left\{ \begin{array}{ll} u_t = u_{xx} + f(u), & x>0, \ t >0,\\ u(0,t) = b u_x(0,t), & t>0,\\ u(x,0) = u_0 (x)\geqslant 0 , & x\geqslant 0, \end{array} \right. $$ where $b\geqslant 0$ and $f$ is an unbalanced bistable nonlinearity. By investigating families of initial data of the type $\{ σϕ\}_{σ>0}$, where $ϕ$ belongs to an appropriate class of nonnegative compactly supported functions, we exhibit the sharp threshold between vanishing and spreading. More specifically, there exists some value $σ^*$ such that the solution converges uniformly to 0 for any $0 < σ< σ^*$, and locally uniformly to a positive stationary state for any $ σ> σ^*$. In the threshold case $σ= σ^*$, the profile of the solution approaches the symmetrically decreasing ground state with some shift, which may be either finite or infinite. In the latter case, the shift evolves as $C \ln t$ where~$C$ is a positive constant we compute explicitly, so that the solution is traveling with a pulse-like shape albeit with an asymptotically zero speed. Depending on $b$, but also in some cases on the choice of the initial datum, we prove that one or both of the situations may happen.

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