Electron. J. Diff. Equ., Vol. 2015 (2015), No. 187, pp. 1-26.

Singular limit solutions for 4-dimensional stationary Kuramoto-Sivashinsky equations with exponential nonlinearity

Sami Baraket, Moufida Khtaifi, Taieb Ouni

Abstract:
Let $\Omega$ be a bounded domain in $\mathbb{R}^4$ with smooth boundary, and let $x_1, x_2, \dots, x_m $ be points in $\Omega$. We are concerned with the singular stationary non-homogenous Kuramoto-Sivashinsky equation
$$ \Delta^2 u -\gamma\Delta u- \lambda|\nabla u|^2 = \rho^4f(u), $$
where f is a function that depends only the spatial variable. We use a nonlinear domain decomposition method to give sufficient conditions for the existence of a positive weak solution satisfying the Dirichlet-like boundary conditions $u =\Delta u =0$, and being singular at each $x_i$ as the parameters $\lambda, \gamma$ and $\rho$ tend to 0. An analogous problem in two-dimensions was considered in [2] under condition (A1) below. However we do not assume that condition.

Submitted January 28, 2015. Published July 13, 2015.
Math Subject Classifications: 58J08, 35J40, 35J60, 35J75.
Key Words: Singular limits; Green's function; Kuramoto-Sivashinsky equation; domain decomposition method.

Show me the PDF file (373 KB), TEX file, and other files for this article.

Sami Baraket
Department of Mathematics, College of Science
King Saud University, P.O. Box 2455
Riyadh 11451, Saudi Arabia
email: sbaraket@ksu.edu.sa
Moufida Khtaifi
Département de Mathématiques
Faculté des Sciences de Tunis Campus Universitaire
Université Tunis Elmanar, 2092 Tunis, Tunisia
email: moufida180888@gmail.com
Taieb Ouni
Département de Mathématiques
Faculté des Sciences de Tunis Campus Universitaire
Université Tunis Elmanar, 2092 Tunis, Tunisia
email: Taieb.Ouni@fst.rnu.tn

Return to the EJDE web page