Summary: In this work we consider equations of the form \[ \partial_t u+P\big(\partial_x \big) u+G\big( u,\partial_x u,\ldots,\partial_x^l u\big)=0, \] where \(P\) is any polynomial without constant term, and \(G\) is any polynomial without constant or linear terms. We prove that if \(u\) is a sufficiently smooth solution of the equation, such that \(\mathrm{supp}\, u(0),\mathrm{supp}\, u(T)\subset (-\infty,B]\) for some \(B>0\), then there exists \(R_0 >0\) such that \(\mathrm{supp}\, u(t)\subset (-\infty, R_0]\) for every \(t\in [0,T]\). Then, as an example of the application of this result, we employ it to show a unique continuation principle for the Kawahara equation, \[ \partial_t u+\partial_x^5 u+\partial_x^3 u+u\partial_x u=0, \] and for the generalized KdV hierarchy \[ \partial_t u+ (-1)^{k+1}\partial_x^{2k+1} u+G\big( u,\partial_x u,\ldots, \partial_x^{2k}u\big) =0. \]
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