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Recently, Gu et al. [ 7 , 8 ] studied a reaction-diffusion-advection equation \begin{document}$u_t =u_{xx} -β u_x + f(u)$\end{document} in \begin{document}$(g(t), h(t))$\end{document} , where \begin{document}$g(t)$\end{document} and \begin{document}$h(t)$\end{document} are two free boundaries satisfying Stefan conditions, \begin{document}$f(u)$\end{document} is a Fisher-KPP type of nonlinearity. When \begin{document}$β ∈ [0,c_0)$\end{document} , where \begin{document}$c_0 := 2\sqrt{f'(0)}$\end{document} , they found that for a spreading solution \begin{document}$(u,g,h)$\end{document} , \begin{document}$h(t)/t \to c^*_r (β)$\end{document} and \begin{document}$g(t)/t \to c^*_l (β)$\end{document} as \begin{document}$t \to ∞$\end{document} , and \begin{document}$c^*_r (β) > c^*_r(0) = - c^*_l (0) > - c^*_l (β) >0$\end{document} . In this paper we study the expanding speed \begin{document}$C^*(β) :=c^*_r(β) - c^*_l (β)$\end{document} of the habitat \begin{document}$(g(t), h(t))$\end{document} , and show that \begin{document}$C^*(β)$\end{document} is strictly increasing in \begin{document}$β ∈ [0,c_0)$\end{document} . When \begin{document}$β ∈ [c_0, β^*)$\end{document} for some \begin{document}$β^*>c_0$\end{document} , [ 8 ] also found a virtual spreading phenomena: \begin{document}$h(t)/t \to c^*_r(β)$\end{document} as \begin{document}$t\to∞$\end{document} , and a back forms in the solution which moves rightward with a speed \begin{document}$β - c_0$\end{document} . Hence the expanding speed of the main habitat for such a solution is \begin{document}$C^*(β) := c^*_r(β) -[β -c_0]$\end{document} . In this paper we show that \begin{document}$C^*(β)$\end{document} is strictly decreasing in \begin{document}$β∈ [c_0, β^*)$\end{document} with \begin{document}$C^*(β^* -0)=0$\end{document} , and so there exists a unique \begin{document}$β_0∈ (c_0, β^*)$\end{document} such that the advection is favorable to the expanding speed of the habitat if and only if \begin{document}$β∈ (0,β_0)$\end{document} . |
