arXiv subjects

Hongkai Cao

Publications and source records attributed to Hongkai Cao.

3 recordsLinked to original sources

arXiv2026

The high dimensional monostable reaction-diffusion equation with free boundary and radial symmetry

We consider the radially symmetric version of the reaction-diffusion equation $u_t-d\Delta u=f(u)$ with a monostable nonlinearity $f$, viewed as a model for the spreading of a species with population range $r 0$ and $h'(t)=-d u_r(t,h(t))/\delta$. For the one-dimensional case ($N=1$), Du \cite{DN} proved that when $\delta\in(0,1)$, spreading occurs: $u\to1$ locally uniformly in $\mathbb{R}$, $h(t)\to\infty$, and $\lim_{t\to\infty}[h(t)-c_*t]=\tilde{h}\in\mathbb{R}$ with no logarithmic shift. In the present paper we consider $N\ge2$ and establish a complete trichotomy: spreading for $\delta\in(0,1)$; transition for $\delta=1$, where $u\to1$ uniformly on $[0,h(t)]$ and $h(t)\to h_\infty\in(0,\infty)$; and vanishing for $\delta>1$, where $h(t)\to0$ and $u\to\delta$ uniformly on $[0,h(t)]$. For the spreading regime, by constructing sharp upper and lower solutions, we prove that the solution converges globally to the semi-wave profile and reveal a logarithmic shift of the form $ \lim_{t\to\infty}\big[h(t)-c_*t+c_N(\delta)\log t\big]=\hat{h}\in\mathbb{R}$, with the coefficient $c_N(\delta)>0$ satisfying $ \lim_{\delta\to0}c_N(\delta)=d(N-1)/c_0$, where $d(N-1)/c_0$ is the shift coefficient for the high-dimensional radial pushed-case Cauchy problem. These results reveal the connection to the spreading behavior modeled by the corresponding Cauchy problem.

math.AP↗

arXiv2026

Trichotomy dynamics of a free boundary model for biological invasion

It is well known that the reaction-diffusion equation $u_t=du_{xx}+f(u)$ with compactly supported nonnegative initial functions exhibits trichotomy dynamics for bistable and combustion type $f(u)$ \cite{DM, zlatos}. The same is true for the corresponding Stefan type free boundary problem \cite{DL}. In this paper, we reveal a rather different type of trichotomy for this reaction-diffusion equation under a new set of (free) boundary conditions, arising as a model for biological invasion with $u(t,x)$ representing the density of an invading species over the one dimensional spatial regin $[0, h(t)]$. The evolution of the invading front $x=h(t)$ is governed by $h'(t)=-\frac d\delta u_x(t, h(t))$ and $u(t, h(t))=\delta\in (\hat\theta_f, 1)$, with $\hat\theta_f \in [0, 1)$ uniquely determined by $f$; they allow $h(t)$ to advance as well as to retreat when time increases. At the fixed boundary $x=0$, the density is controlled by $u(t,0)=\delta_0\geq 0$. We completely classify the long-time dynamics of the model when $f(u)$ is a monostable, or bistable, or combustion type nonlinear function. In the biologically interesting case that $\delta_0<\delta$, we show that there are exactly three scenarios: (i) successful spreading, (ii) finite-time vanishing, (iii) a transition state characterized by $h(t)\to l_*\in (0, \infty)$ and $u(t,x)\to w_*(x)$ as $t\to\infty$, where $(u(t,x), h(t))\equiv (w_*(x), l_*)$ is the unique stationary solution of the free boundary problem. The model here does not have the usual order-preserving property enjoyed by those considered in \cite{DM, zlatos, DL} and elsewhere (i.e., $u(0,x)\leq v(0,x)$ implies $u(t,x)\leq v(t,x)$ for all $t>0$ if $u$ and $v$ are two solutions of the problem), which is intrinsically linked to the many novel features of the model.

math.AP↗

arXiv2025

Convergence to a receding wave in a monostable free boundary problem

We study a monostable reaction-diffusion equation of the form $u_t=du_{xx}+f(u)$ over a semi-infinite spatial domain $[g(t),\infty)$, with $x=g(t)$ the free boundary whose evolution is governed by equations derived from a ``preferred population density'' principle, which postulates that the species with population density $u(t,x)$ and population range $[g(t),\infty)$ maintains a certain density $\delta$ at the habitat edge $x=g(t)$. In the ``high-density'' regime, where $\delta$ exceeds the carrying capacity of the favourable environment represented by a monostable function $f(u)$, it is known (see \cite{DLNS} for the case of a bounded population range $[g(t), h(t)]$) that for large time, the front retreats as time advances. In this work, the unboundedness of the population range $[g(t),\infty)$ allows us to prove that, as time $t$ converges to infinity, the free boundary $x=g(t)$ converges to $\infty$ with a constant asymptotic speed $c(\delta)>0$ determined by an associated semi-wave problem, and the population density $u(t,x)$ has the property that $u(t,x+g(t))$ converges uniformly to $q_{c(\delta)}(x)$, the semi-wave profile function associated with the speed $c(\delta)$. It turns out that in the retreating situation considered here, some key techniques developed for advancing fronts in related free boundary models do not work anymore. This difficulty is overcome here by a ``touching method", which uses a family of lower and upper solutions constructed from semi-waves of some carefully designed auxiliary problems to touch the solution $u(t,x)$ at the moving boundary $x=g(t)$, thereby generating a setting where the comparison principle can be used to obtain the desired estimates for $g'(t)$ and $u(t,x)$. We believe this method will find applications elsewhere.

math.AP↗