arXiv subjects
Publications and source records attributed to Bendong Lou.
In this note, we propose Bernstein's problem and De Giorgi's conjecture for spatially inhomogeneous equations, as well as De Giorgi's conjecture for system of reaction-diffusion equations.
We study a nonlinear diffusion equation of the form $u_t=u_{xx}+f(u)\ (x\in [g(t),h(t)])$ with free boundary conditions $g'(t)=-u_x(t,g(t))+α$ and $h'(t)=-u_x(t,g(t))-α$ for some $α>0$. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundaries representing the expanding fronts. When $α=0$, the problem was recently investigated by \cite{DuLin, DuLou}. In this paper we consider the case $α>0$. In this case shrinking (i.e. $h(t)-g(t)\to 0$) may happen, which is quite different from the case $α=0$. Moreover, we show that, under certain conditions on $f$, shrinking is equivalent to vanishing (i.e. $u\to 0$), both of them happen as $t$ tends to some finite time. On the other hand, every bounded and positive time-global solution converges to a nonzero stationary solution as $t\to \infty$. As applications, we consider monostable and bistable types of nonlinearities, and obtain a complete description on the asymptotic behavior of the solutions.
We study a two-point free boundary problem in a sector for a quasilinear parabolic equation. The boundary conditions are assumed to be spatially and temporally "self-similar" in a special way. We prove the existence, uniqueness and asymptotic stability of an expanding solution which is self-similar at discrete times. We also study the existence and uniqueness of a shrinking solution which is self-similar at discrete times.
We study two systems of reaction diffusion equations with monostable or bistable type of nonlinearity and with free boundaries. These systems are used as multi-species competitive model. For two-species models, we prove the existence of a traveling wave solution which consists of two semi-waves intersecting at the free boundary. For three-species models, we also prove the existence of a traveling wave solution which, however, consists of two semi-waves and one compactly supported wave in between, each intersecting with its neighbor at the free boundary.
In this paper, we consider a Fisher-KPP equation with an advection term and two free boundaries, which models the behavior of an invasive species in one dimension space. When spreading happens (that is, the solution converges to a positive constant), we use phase plane analysis and upper/lower solutions to prove that the rightward and leftward asymptotic spreading speeds exist, both are positive constants. Moreover, one of them is bigger and the other is smaller than the spreading speed in the corresponding problem without advection term.
We study nonlinear diffusion problems of the form $u_t=u_{xx}+f(u)$ with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For special $f(u)$ of the Fisher-KPP type, the problem was investigated by Du and Lin [8]. Here we consider much more general nonlinear terms. For any $f(u)$ which is $C^1$ and satisfies $f(0)=0$, we show that the omega limit set $ω(u)$ of every bounded positive solution is determined by a stationary solution. For monostable, bistable and combustion types of nonlinearities, we obtain a rather complete description of the long-time dynamical behavior of the problem; moreover, by introducing a parameter $σ$ in the initial data, we reveal a threshold value $σ^*$ such that spreading ($\lim_{t\to\infty}u= 1$) happens when $σ>σ^*$, vanishing ($\lim_{t\to\infty}u=0$) happens when $σ<σ^*$, and at the threshold value $σ^*$, $ω(u)$ is different for the three different types of nonlinearities. When spreading happens, we make use of "semi-waves" to determine the asymptotic spreading speed of the front.