 Solution

The algebraic change of variables still works for $a>0$, but the transformed initial data are now multiplied by $e^{aX/3}$. Mere decay of $Q_0$ does not ensure existence of the Half-range Fourier transform used above. For example, $Q_0(X)=e^{-dX}$ with $0<d<a/3$ gives a growing transformed profile.

There is also a genuine stability distinction. On the whole line a real-frequency mode of the original equation evolves with factorIts modulus is $e^{(a{K}^2-c)T}$, so positive $a$ amplifies arbitrarily high frequencies rather than damping them. Thus the formal gauge equivalence does not justify the same unweighted Sobolev well-posedness conclusion. Weighted or more strongly decaying data and corresponding contour changes may allow particular solutions; the assumptions stated for the negative-$a$ construction cannot simply be reused.