The Korteweg-de Vries equation in classes of increasing functions with prescribed asymptotics as $|x|\to\infty$

The Cauchy problem is considered for the Korteweg–de Vries equation with an increasing initial function admitting an asymptotic expansion in decreasing powers of $x$ as $|x|\to\infty$. It is proved that asymptotic solutions having the form of series in decreasing powers of $x$ differ from the actual solutions by a function $w(x,t)$ smooth in $t$ with values in $S(\mathbf R_x)$.
Bibliography: 3 titles.