We study a free boundary problem for Fisher-KPP equation: ut=uxx+f(u)\documentclass[12pt]{minimal}
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\begin{document}$$u_t=u_{xx}+f(u)$$\end{document} (g(t)<x<h(t)\documentclass[12pt]{minimal}
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\begin{document}$$g(t)< x < h(t)$$\end{document}) with free boundary conditions h′(t)=-ux(t,h(t))-β\documentclass[12pt]{minimal}
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\begin{document}$$h'(t)=-u_x(t,h(t))-\beta $$\end{document} and g′(t)=-ux(t,g(t))-α\documentclass[12pt]{minimal}
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\begin{document}$$g'(t)=-u_x(t,g(t))-\alpha $$\end{document} for α>0\documentclass[12pt]{minimal}
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\begin{document}$$\alpha >0$$\end{document} and β∈R\documentclass[12pt]{minimal}
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\begin{document}$$\beta \in \mathbb {R}$$\end{document}. Such a free boundary problem can model the spreading of a biological or chemical species affected by the boundary environment. β>0\documentclass[12pt]{minimal}
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\begin{document}$$\beta >0$$\end{document} means that there is a “resistance force” with strength β\documentclass[12pt]{minimal}
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\begin{document}$$\beta $$\end{document} at boundary x=h(t)\documentclass[12pt]{minimal}
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\begin{document}$$x=h(t)$$\end{document}. β<0\documentclass[12pt]{minimal}
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\begin{document}$$\beta <0$$\end{document} (resp. α>0\documentclass[12pt]{minimal}
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\begin{document}$$\alpha >0$$\end{document}) means that there is an enhancing force with strength β\documentclass[12pt]{minimal}
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\begin{document}$$\beta $$\end{document} (resp. α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}) at the boundary x=h(t)\documentclass[12pt]{minimal}
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\begin{document}$$x=h(t)$$\end{document} (resp. g(t)). There are many parts of (α,β)\documentclass[12pt]{minimal}
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\begin{document}$$(\alpha ,\beta )$$\end{document}. 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 (α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} or β\documentclass[12pt]{minimal}
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\begin{document}$$\beta $$\end{document} large), any solution will vanish, also there are three types of vanishing. For the case α=β\documentclass[12pt]{minimal}
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\begin{document}$$\alpha = \beta $$\end{document}, we have two different trichotomy results and a dichotomy result.
