We study a free boundary problem for Fisher-KPP equation: \(u_t=u_{xx}+f(u)\) (\(g(t)< x < h(t)\)) with free boundary conditions \(h'(t)=-u_x(t,h(t))-\beta \) and \(g'(t)=-u_x(t,g(t))-\alpha \) for \(\alpha >0\) and \(\beta \in \mathbb {R}\). Such a free boundary problem can model the spreading of a biological or chemical species affected by the boundary environment. \(\beta >0\) means that there is a “resistance force” with strength \(\beta \) at boundary \(x=h(t)\). \(\beta <0\) (resp. \(\alpha >0\)) means that there is an enhancing force with strength \(\beta \) (resp. \(\alpha \)) at the boundary \(x=h(t)\) (resp. g(t)). There are many parts of \((\alpha ,\beta )\). In different parts, the asymptotic behavior of solutions are different. In the first part, we have a spreading-transition-vanishing result: either spreading happens (the solution converges to 1 in the moving frame), or in the transition case (the solution will converge to the compactly supported traveling wave), or vanishing happens (the solution converges to 0 within a finite time). In the second part, we also have a trichotomy result, but in transition case the solution will converge to the non-monotonous traveling semi-wave, and the vanishing case has three different types. For the third part, only spreading happens for any solution. In the fourth part (\(\alpha \) or \(\beta \) large), any solution will vanish, also there are three types of vanishing. For the case \(\alpha = \beta \), we have two different trichotomy results and a dichotomy result.

我们研究了Fisher-KPP方程的自由边界问题：\（u_t = u_ {xx} + f（u）\）（\（g（t）<x <h（t）\））具有自由边界条件\（ h'（t）=-u_x（t，h（t））-\ beta \）和\（g'（t）=-u_x（t，g（t））-\ alpha \）对于\（\ alpha > 0 \）和\（\ beta \ in \ mathbb {R} \）中。这种自由边界问题可以模拟受边界环境影响的生物或化学物种的扩散。\（\ beta> 0 \）表示在边界\（x = h（t）\）处具有强度为\（\ beta \）的“阻力” 。\（\ beta <0 \）（分别是\（\ alpha> 0 \））意味着存在强度增强的力量\（\ beta \）（分别是\（\ alpha \））在边界\（x = h（t）\）（分别是g（t））处。\（（（\ alpha，\ beta）\）有很多部分。在不同的部分，解的渐近行为是不同的。在第一部分中，我们有一个扩散过渡消失的结果：要么发生扩散（解在移动框架中收敛到1），要么在过渡情况下（该解决方案收敛到紧凑支撑的行波），或者消失发生（解决方案在有限时间内收敛到0）。在第二部分中，我们也有三分法的结果，但是在过渡情况下，解决方案将收敛到非单调行进半波，而消失情况有三种不同类型。第三部分，任何解决方案都只会传播。在第四部分（\（\ alpha \）或\（\ beta \）大）中，任何解决方案都将消失，也有三种消失类型。对于这种情况\（\ alpha = \ beta \），我们有两个不同的三分法结果和二分法结果。