Properties of the support of solutions of a class of nonlinear evolution equations

In this work we consider equations of the form ∂tu+P(∂x)u+G(u,∂xu,⋯,∂xlu)=0,$$\begin{equation*}\hskip7pc \partial _t u+P\big (\partial _x\big ) u+G\big (u,\partial _xu,\dots ,\partial _x^l u\big )=0, \end{equation*}$$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 suppu(0),suppu(T)⊂(−∞,B]$\operatorname{supp}u(0),\operatorname{supp}u(T)\subset { (-\infty ,B ]}$ for some B>0$B>0$, then there exists R0>0$R_0>0$ such that suppu(t)⊂(−∞,R0]$\operatorname{supp}u(t)\subset (-\infty ,R_0]$ for every t∈[0,T]$t\in [0,T]$. Then, as an example of the application of this result, we employ it to show a unique continuation principle for the Kawahara equation, ∂tu+∂x5u+∂x3u+u∂xu=0,$$\begin{equation*}\hskip9pc \partial _t u+\partial _x^5 u+\partial _x^3 u+u\partial _x u=0, \end{equation*}$$and for the generalized KdV hierarchy ∂tu+(−1)k+1∂x2k+1u+G(u,∂xu,⋯,∂x2ku)=0.$$\begin{equation*}\hskip6pc \partial _t u+ (-1)^{k+1}\partial _x^{2k+1} u+G\big (u,\partial _x u,\dots , \partial _x^{2k}u\big ) =0. \end{equation*}$$