Time decay estimates for solutions of the Cauchy problem for the modified Kawahara equation

The large-time behaviour of solutions of the Cauchy problem for the modified Kawahara equation
$$ \begin{cases} u_t-\partial_xu^3-\frac a3\partial_x^3u+\frac b5\partial_x^5u=0,&(t,x)\in\mathbb R^2,\\ u(0,x)=u_0(x),&x\in\mathbb R, \end{cases} $$
where $a,b>0$, is investigated. Under the assumptions that the total mass of the initial data $\int u_0(x)\,dx$ is nonzero and the initial data $u_0$ are small in the norm of $\mathbf H^{2,1}$ it is proved that a global-in-time solution exists and estimates for its large-time decay are found.
Bibliography: 19 titles.