arXiv 1406.46292014-06-18math.AP

具有自由边界条件的反应扩散方程解的渐近性

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

Jingjing Cai, Bendong Lou, Maolin Zhou

备注：: 17 pages
DOI：: 10.1007/s10884-014-9404-z

摘要：

研究了一类具有自由边界条件$g'(t)=-u_x(t,g(t))+α$和$h'(t)=-u_x(t,g(t))-α$的$u_t=u_{xx}+f(u)\ (x\in [g(t),h(t)])$型非线性扩散方程，其中对某些$α>0$ .这些问题可以用来描述生物或化学物种的扩散，自由边界代表扩展的前沿。当$α=0$时，\cite{DuLin, DuLou}最近研究了该问题。在本文中，我们考虑的情况下$α>0$。在这种情况下，可能会发生收缩（即$h(t)-g(t)\to 0$），这与$α=0$的情况完全不同。进一步证明了在一定条件下，$f$上的收缩等价于消失（即$u\to 0$），它们都发生在$t$趋于某个有限时间时.另一方面，每一个有界的正时间全局解收敛到一个非零平稳解，如$t\to \infty$。作为应用，我们考虑了单稳态和双稳态的非线性项，得到了解的渐近性态的完整描述。

英文摘要：

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.

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摘要：

英文摘要：

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