We consider the semilinear parabolic equation $u_t=u_{xx}+f(u)$ on the realline, where $f$ is a locally Lipschitz function on $\mathbb{R}.$ We prove thatif a solution $u$ of this equation is bounded and its initial value $u(x,0)$has distinct limits at $x=\pm\infty,$ then the solution is quasiconvergent,that is, all its limit profiles as $t\to\infty$ are steady states.
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机译：我们考虑在实线上的半线性抛物线方程$ u_t = u_ {xx} + f（u）$，其中$ f $是$ \ mathbb {R}上的局部Lipschitz函数。该方程是有界的，其初始值$ u（x，0）$在$ x = \ pm \ infty，$处具有明显的极限，则解是准收敛的，即所有极限轮廓为$ t \ to \ infty $处于稳定状态。
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