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 ${\theta }^{\pm }$ 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: ${\theta }^{\pm }={\theta }_0$. 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 $f({\theta }_0)\ne 0$, or $f({\theta }_0)=0$ and ${\theta }_0$ is a stable equilibrium of the equation $\dot{\xi }=f(\xi )$. If $f({\theta }_0)=0$ and ${\theta }_0$ is an unstable equilibrium of the equation $\dot{\xi }=f(\xi )$, 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.

我们继续研究半线性抛物线方程的有界解 ${你}_{吨}={你}_{XX}+F(你)$在实线上，其中f是局部 Lipschitz 函数$\mathbb{电阻}$. 假设初始值${你}_0=你(\cdot ,0)$ 解的极限是有限的 ${\theta }^{\pm }$ 作为 $X\to \pm \infty$，我们的目标是描述渐近行为 $你(X,吨)$ 作为 $吨\to \infty$. 在之前的工作中，我们证明了如果两个极限不同，则解是拟收敛的，即其所有局部均匀极限剖面为$吨\to \infty$是稳态。众所周知，如果限制相等，则此结果通常无效：${\theta }^{\pm }={\theta }_0$. 在本文中，我们仔细研究了等限情况。在非线性的次要非简并假设下，我们证明了解是拟收敛的，如果$F({\theta }_0)\ne 0$， 要么 $F({\theta }_0)=0$ 和 ${\theta }_0$ 是方程的稳定平衡 $\dot{\xi }=F(\xi )$. 如果$F({\theta }_0)=0$ 和 ${\theta }_0$ 是方程的不稳定平衡 $\dot{\xi }=F(\xi )$，我们还证明了一些准收敛定理（必要地）对 ${你}_0$. 我们证明拟收敛定理的一个主要成分——以及独立兴趣的结果——是将某种类型的整个解决方案分类为稳态和两个不相交的稳态集之间的异宿连接。