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

P. I. Naumkin

Sbornik. Mathematics, Tome 210 (2019) no. 5, pp. 693-730

Résumé

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.

Détail

Keywords: Kawahara equation, cubic nonlinearity, large-time asymptotics.

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