 Nonlinear endpoint layer with a quadratic reaction

For $q>0$, matching $Y\to -q$, $Y^{\prime }\to 0$ fixes the first integral to $(Y^{\prime }{)}^2=(q-2Y)(Y+q{)}^2/3$. Hence endpoint values above $q/2$ cannot match this outer branch. The upper family is $Y=-q+(3q/2){\mathrm{sech}}^2[\sqrt{q}(\xi -{\xi }_0)/2]$. An endpoint value strictly between $-q$ and $q/2$ admits two translations, one initially approaching $-q$ and one initially moving towards the turning value $q/2$. Values below $-q$ give the unique lower family $Y=-q-(3q/2){\mathrm{csch}}^2[\sqrt{q}(\xi +{\xi }_0)/2]$ with ${\xi }_0>0$. Matching to zero instead gives $(Y^{\prime }{)}^2=-q{Y}^2-2Y^3/3$, which forbids a nonconstant real approach to zero. These are leading boundary layer classifications, not exact multiplicity theorems at degenerate endpoint data.