Electronic Journal of Differential Equations,
Vol. 2015 (2015), No. 187, pp. 1-26.

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

Authors: Sami Baraket  (King Saud Univ., Riyadh, Saudi Arabia)
         Moufida Khtaifi (Univ. Tunis Elmanar, Tunis, Tunisia)
         Taieb Ouni  (Univ. Tunis Elmanar, Tunis, Tunisia)

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.