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.