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 \(x^{1}, x^{2}, \ldots, x^{m}\) 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.