We consider the first initial-boundary value problem for the 2-D Kuramoto-Sivashinsky equation in a unit disk with homogeneous boundary conditions, periodicity conditions in the angle, and small initial data. Apart from proving the existence and uniqueness of a global in time solution, we construct it in the form of a series in a small parameter present in the initial conditions. In the stable case we also obtain the uniform in space long-time asymptotic expansion of the constructed solution and its asymptotics with respect to the nonlinearity constant. The method can work for other dissipative parabolic equations with dispersion.
@article{bwmeta1.element.bwnjournal-article-apmv73z3p227bwm,
author = {Varlamov, Vladimir},
title = {On the Kuramoto-Sivashinsky equation in a disk},
journal = {Annales Polonici Mathematici},
volume = {75},
year = {2000},
pages = {227-256},
zbl = {0963.35035},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv73z3p227bwm}
}
Varlamov, Vladimir. On the Kuramoto-Sivashinsky equation in a disk. Annales Polonici Mathematici, Tome 75 (2000) pp. 227-256. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv73z3p227bwm/
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