Table 2.
Open in new tab

|$ \hat{u}_{0} \lt 0 $|⁠: The parameters satisfy the inequalities: |$ 0\,\lt\,c^{*} \lt -\hat{u}_{0} \lt\tilde{u}_{0} \lt\hat{u}_{0}+2\sigma. $|

Region Most Dominant term Asymptotics
dominant in |$ E $|
terms
|$ 0\,\lt\,y\,\lt\,c^{*} $||$ \hat{J}_{0}\gg I_{0}\gg $||$ L_{0}/\hat{J}_{0}= $||$ U_{{\rm tw}}+E $|
|$ L_{0}\gg I_{1} $||$ ~{}O\left(\mathrm{e}^{c^{*}t(y-k^{*})}\right) $|
|$ c^{*} \lt y \lt $||$ I_{0}\gg\hat{J}_{0}\gg $||$ L_{0}/I_{0}= $||$ U_{{\rm tw}}+E $|
|$ \quad\quad-\hat{u}_{0} $||$ L_{0}\gg I_{1} $||$ ~{}O\left(\mathrm{e}^{\hat{u}_{0}ty}\right) $|
|$ -\hat{u}_{0} \lt y \lt $||$ I_{0}\gg\hat{J}_{0}\gg $||$ I_{1}/I_{0}= $||$ U_{{\rm tw}}+E $|
|$ \quad\quad\tilde{u}_{0} $||$ ~{}~{}I_{1}=O(L_{0}) $||$ O\left(t^{-1/2}\mathrm{e}^{-(y-\hat{u}_{0})^{2}t/4}\right) $|
|$ \tilde{u}_{0} \lt y \lt $||$ I_{0}\gg I_{1}= $||$ I_{1}/I_{0}= $||$ u=\hat{u}_{0}+ $|
|$ \hat{u}_{0}+2\sigma $||$ O(\hat{J}_{0})=O(L_{0}) $||$ O\left(t^{-1/2}\mathrm{e}^{-(y-\hat{u}_{0})^{2}t/4}\right) $||$ O\left(t^{-1/2}\mathrm{e}^{-(y-\hat{u}_{0})^{2}t/4}\right) $|
|$ y \gt\hat{u}_{0}+2\sigma $||$ I_{0}\gg I_{1}\gg $||$ I_{1}/I_{0}= $||$ u=\hat{u}_{0}+u_{\infty}\times $|
|$ \hat{J}_{0}=O(L_{0}) $||$ ~{}O\left(\mathrm{e}^{-\sigma t(y-\sigma-\hat{u}_{0})}\right) $||$ \mathrm{e}^{-\sigma t(y-\sigma-\hat{u}_{0})}+o(1) $|
Region Most Dominant term Asymptotics
dominant in |$ E $|
terms
|$ 0\,\lt\,y\,\lt\,c^{*} $||$ \hat{J}_{0}\gg I_{0}\gg $||$ L_{0}/\hat{J}_{0}= $||$ U_{{\rm tw}}+E $|
|$ L_{0}\gg I_{1} $||$ ~{}O\left(\mathrm{e}^{c^{*}t(y-k^{*})}\right) $|
|$ c^{*} \lt y \lt $||$ I_{0}\gg\hat{J}_{0}\gg $||$ L_{0}/I_{0}= $||$ U_{{\rm tw}}+E $|
|$ \quad\quad-\hat{u}_{0} $||$ L_{0}\gg I_{1} $||$ ~{}O\left(\mathrm{e}^{\hat{u}_{0}ty}\right) $|
|$ -\hat{u}_{0} \lt y \lt $||$ I_{0}\gg\hat{J}_{0}\gg $||$ I_{1}/I_{0}= $||$ U_{{\rm tw}}+E $|
|$ \quad\quad\tilde{u}_{0} $||$ ~{}~{}I_{1}=O(L_{0}) $||$ O\left(t^{-1/2}\mathrm{e}^{-(y-\hat{u}_{0})^{2}t/4}\right) $|
|$ \tilde{u}_{0} \lt y \lt $||$ I_{0}\gg I_{1}= $||$ I_{1}/I_{0}= $||$ u=\hat{u}_{0}+ $|
|$ \hat{u}_{0}+2\sigma $||$ O(\hat{J}_{0})=O(L_{0}) $||$ O\left(t^{-1/2}\mathrm{e}^{-(y-\hat{u}_{0})^{2}t/4}\right) $||$ O\left(t^{-1/2}\mathrm{e}^{-(y-\hat{u}_{0})^{2}t/4}\right) $|
|$ y \gt\hat{u}_{0}+2\sigma $||$ I_{0}\gg I_{1}\gg $||$ I_{1}/I_{0}= $||$ u=\hat{u}_{0}+u_{\infty}\times $|
|$ \hat{J}_{0}=O(L_{0}) $||$ ~{}O\left(\mathrm{e}^{-\sigma t(y-\sigma-\hat{u}_{0})}\right) $||$ \mathrm{e}^{-\sigma t(y-\sigma-\hat{u}_{0})}+o(1) $|
Table 2.
Open in new tab

|$ \hat{u}_{0} \lt 0 $|⁠: The parameters satisfy the inequalities: |$ 0\,\lt\,c^{*} \lt -\hat{u}_{0} \lt\tilde{u}_{0} \lt\hat{u}_{0}+2\sigma. $|

Region Most Dominant term Asymptotics
dominant in |$ E $|
terms
|$ 0\,\lt\,y\,\lt\,c^{*} $||$ \hat{J}_{0}\gg I_{0}\gg $||$ L_{0}/\hat{J}_{0}= $||$ U_{{\rm tw}}+E $|
|$ L_{0}\gg I_{1} $||$ ~{}O\left(\mathrm{e}^{c^{*}t(y-k^{*})}\right) $|
|$ c^{*} \lt y \lt $||$ I_{0}\gg\hat{J}_{0}\gg $||$ L_{0}/I_{0}= $||$ U_{{\rm tw}}+E $|
|$ \quad\quad-\hat{u}_{0} $||$ L_{0}\gg I_{1} $||$ ~{}O\left(\mathrm{e}^{\hat{u}_{0}ty}\right) $|
|$ -\hat{u}_{0} \lt y \lt $||$ I_{0}\gg\hat{J}_{0}\gg $||$ I_{1}/I_{0}= $||$ U_{{\rm tw}}+E $|
|$ \quad\quad\tilde{u}_{0} $||$ ~{}~{}I_{1}=O(L_{0}) $||$ O\left(t^{-1/2}\mathrm{e}^{-(y-\hat{u}_{0})^{2}t/4}\right) $|
|$ \tilde{u}_{0} \lt y \lt $||$ I_{0}\gg I_{1}= $||$ I_{1}/I_{0}= $||$ u=\hat{u}_{0}+ $|
|$ \hat{u}_{0}+2\sigma $||$ O(\hat{J}_{0})=O(L_{0}) $||$ O\left(t^{-1/2}\mathrm{e}^{-(y-\hat{u}_{0})^{2}t/4}\right) $||$ O\left(t^{-1/2}\mathrm{e}^{-(y-\hat{u}_{0})^{2}t/4}\right) $|
|$ y \gt\hat{u}_{0}+2\sigma $||$ I_{0}\gg I_{1}\gg $||$ I_{1}/I_{0}= $||$ u=\hat{u}_{0}+u_{\infty}\times $|
|$ \hat{J}_{0}=O(L_{0}) $||$ ~{}O\left(\mathrm{e}^{-\sigma t(y-\sigma-\hat{u}_{0})}\right) $||$ \mathrm{e}^{-\sigma t(y-\sigma-\hat{u}_{0})}+o(1) $|
Region Most Dominant term Asymptotics
dominant in |$ E $|
terms
|$ 0\,\lt\,y\,\lt\,c^{*} $||$ \hat{J}_{0}\gg I_{0}\gg $||$ L_{0}/\hat{J}_{0}= $||$ U_{{\rm tw}}+E $|
|$ L_{0}\gg I_{1} $||$ ~{}O\left(\mathrm{e}^{c^{*}t(y-k^{*})}\right) $|
|$ c^{*} \lt y \lt $||$ I_{0}\gg\hat{J}_{0}\gg $||$ L_{0}/I_{0}= $||$ U_{{\rm tw}}+E $|
|$ \quad\quad-\hat{u}_{0} $||$ L_{0}\gg I_{1} $||$ ~{}O\left(\mathrm{e}^{\hat{u}_{0}ty}\right) $|
|$ -\hat{u}_{0} \lt y \lt $||$ I_{0}\gg\hat{J}_{0}\gg $||$ I_{1}/I_{0}= $||$ U_{{\rm tw}}+E $|
|$ \quad\quad\tilde{u}_{0} $||$ ~{}~{}I_{1}=O(L_{0}) $||$ O\left(t^{-1/2}\mathrm{e}^{-(y-\hat{u}_{0})^{2}t/4}\right) $|
|$ \tilde{u}_{0} \lt y \lt $||$ I_{0}\gg I_{1}= $||$ I_{1}/I_{0}= $||$ u=\hat{u}_{0}+ $|
|$ \hat{u}_{0}+2\sigma $||$ O(\hat{J}_{0})=O(L_{0}) $||$ O\left(t^{-1/2}\mathrm{e}^{-(y-\hat{u}_{0})^{2}t/4}\right) $||$ O\left(t^{-1/2}\mathrm{e}^{-(y-\hat{u}_{0})^{2}t/4}\right) $|
|$ y \gt\hat{u}_{0}+2\sigma $||$ I_{0}\gg I_{1}\gg $||$ I_{1}/I_{0}= $||$ u=\hat{u}_{0}+u_{\infty}\times $|
|$ \hat{J}_{0}=O(L_{0}) $||$ ~{}O\left(\mathrm{e}^{-\sigma t(y-\sigma-\hat{u}_{0})}\right) $||$ \mathrm{e}^{-\sigma t(y-\sigma-\hat{u}_{0})}+o(1) $|
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