Geodesic
Time decay estimates for solutions of the Cauchy problem for the modified Kawahara equation
Sbornik. Mathematics, Tome 210 (2019) no. 5, pp. 693-730

Voir la notice de l'article provenant de la source Math-Net.Ru

The large-time behaviour of solutions of the Cauchy problem for the modified Kawahara equation $$ \begin{cases} u_t-\partial_xu^3-\frac a3\partial_x^3u+\frac b5\partial_x^5u=0,(t,x)\in\mathbb R^2,\\ u(0,x)=u_0(x),\in\mathbb R, \end{cases} $$ 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.
Keywords: Kawahara equation, cubic nonlinearity, large-time asymptotics. 
@article{SM_2019_210_5_a2,
     author = {P. I. Naumkin},
     title = {Time decay estimates for solutions of the {Cauchy} problem for the modified {Kawahara} equation},
     journal = {Sbornik. Mathematics},
     pages = {693--730},
     publisher = {mathdoc},
     volume = {210},
     number = {5},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2019_210_5_a2/}
}
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P. I. Naumkin. Time decay estimates for solutions of the Cauchy problem for the modified Kawahara equation. Sbornik. Mathematics, Tome 210 (2019) no. 5, pp. 693-730. http://geodesic.mathdoc.fr/item/SM_2019_210_5_a2/