arXiv subjects

Maolin Zhou

Publications and source records attributed to Maolin Zhou.

40 records · Page 3Linked to original sources

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.

math.AP↗

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.

math.AP↗

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.

math.AP↗

arXiv2014

Asymptotic behavior of solutions of a reaction diffusion equation with free boundary conditions

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.

math.AP↗