Table 1.
Open in new tab

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

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

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

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