论文标题
与收敛初始数据II：无限限制在真实线上的抛物线方程解决方案解决方案的大型行为
Large-time behavior of solutions of parabolic equations on the real line with convergent initial data II: equal limits at infinity
论文作者
论文摘要
我们继续研究半线性抛物线方程的有界解决方案$ u_t = u_ {xx}+f（u）$在实际线上，其中$ f $是$ \ m m i \ mathbb {r}上的本地lipschitz函数。 $ x \ to \ pm \ infty $，我们的目标是将$ u（x，t）$的渐近行为描述为$ t \ to \ infty $。在先前的工作中，我们表明，如果两个限制是不同的，那么解决方案是准ver剂，也就是说，其所有局部统一的限制概况为$ t \ to \ to \ infty $都是稳态。众所周知，如果限制相等，则该结果通常无效：$θ^\ pm =θ_0$。在本文中，我们仔细观察了相等的限制情况。在对非线性的较小非统一假设下，我们表明，如果$ f（θ_0）\ ne0 $或$ f（θ_0）= 0 $和$θ_0$是等式$ \ dot $ \ dot $ \ dot之一。如果$ f（θ_0）= 0 $和$θ_0$是方程式$ \ dotξ= f（ξ）$的不稳定平衡，我们还证明了对$ u_0 $的一些准定理定理（一定是必不可少）。我们证明了准定理的一个主要成分 - 以及独立兴趣的结果----将某种类型的整个溶液分类为稳态和异晶连接之间的整个溶液之间的分类。
We continue our study of bounded solutions of the semilinear parabolic equation $u_t=u_{xx}+f(u)$ on the real line, where $f$ is a locally Lipschitz function on $\mathbb{R}.$ Assuming that the initial value $u_0=u(\cdot,0)$ of the solution has finite limits $θ^\pm$ as $x\to\pm\infty$, our goal is to describe the asymptotic behavior of $u(x,t)$ as $t\to\infty$. In a prior work, we showed that if the two limits are distinct, then the solution is quasiconvergent, that is, all its locally uniform limit profiles as $t\to\infty$ are steady states. It is known that this result is not valid in general if the limits are equal: $θ^\pm=θ_0$. In the present paper, we have a closer look at the equal-limits case. Under minor non-degeneracy assumptions on the nonlinearity, we show that the solution is quasiconvergent if either $f(θ_0)\ne0$, or $f(θ_0)=0$ and $θ_0$ is a stable equilibrium of the equation $\dot ξ=f(ξ)$. If $f(θ_0)=0$ and $θ_0$ is an unstable equilibrium of the equation $\dot ξ=f(ξ)$, we also prove some quasiconvergence theorem making (necessarily) additional assumptions on $u_0$. A major ingredient of our proofs of the quasiconvergence theorems---and a result of independent interest---is the classification of entire solutions of a certain type as steady states and heteroclinic connections between two disjoint sets of steady states.
