= Solution
For every $p<5$, the wave $Q_c$ has <Orbital stability of a generalized KdV solitary wave> in $H^1(\mathbb R)$ modulo translation: for every $\varepsilon>0$ there is $\delta>0$ such that
$$
\|u_0-Q_c\|_{H^1}<\delta
\quad\Longrightarrow\quad
\sup_t\inf_{y\in\mathbb R}
\|u(t)-Q_c(\,\cdot-y)\|_{H^1}<\varepsilon.
$$
The relevant stability slope has the correct sign because scaling gives
$$
\|Q_c\|_2^2
=c^{2/(p-1)-1/2}\|Q\|_2^2
=c^{(5-p)/(2(p-1))}\|Q\|_2^2,
$$
which is strictly increasing in $c$ precisely for $p<5$. Together with the constrained variational characterization of $Q_c$, the conserved mass and energy provide a coercive <Lyapunov function> transverse to the translation direction.
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