On the blowing up solutions of the 4-d general q-Kuramoto-Sivashinsky equation with exponentially »dominated« nonlinearity and singular weight

Abstract

Let $\Omega$ be a bounded domain in $\mathbb{R}^{4}$ with smooth boundary and let $x^{1},x^{2},\dots,x^{m}$ be $m$-points in $\Omega$. We are concerned with the problem $\Delta^{2} u - H(x,u,D^{k}u) = \rho^{4}\prod_{i=1}^{n}|x-p_{i}|^{4\alpha_{i}}f(x)g(u),$ where the principal term is the bi-Laplacian operator, $H(x,u,D^{k}u)$ is a functional which grows with respect to $Du$ at most like $|Du|^{q}$, $1 \leq q \leq 4$, $f:\Omega\to [0,+\infty[$ is a smooth function satisfying $f(p_{i}) \gt 0$ for any $i = 1,\ldots, n$, $\alpha_{i}$ are positives numbers and $g:\mathbb{R} \to [0,+\infty[$ satisfy $|g(u)|\leq ce^{u}$. In this paper, we give sufficient conditions for existence of a family of positive weak solutions $(u_\rho){\rho\gt 0}$ in $\Omega$ under Navier boundary conditions $u=\Delta u =0$ on $\partial \Omega$. The solutions we constructed are singular as the parameters $ho$ tends to 0, when the set of concentration $S={x^{1},\ldots,x^{m}}\subset\Omega$ and the set $\Lambda :={p{1},\ldots, p_{n}}\subset\Omega$ are not necessarily disjoint. The proof is mainly based on nonlinear domain decomposition method.