Opuscula Math. 43, no. 1 (2023), 5-18
https://doi.org/10.7494/OpMath.2023.43.1.5

 
Opuscula Mathematica

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

Sami Baraket
Safia Mahdaoui
Taieb Ouni

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.

Keywords: singular limits, Green's function, nonlinearity, gradient, nonlinear domain decomposition method.

Mathematics Subject Classification: 35J40, 35J08, 35J60, 35J75, 29M27.

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  • Sami Baraket (corresponding author)
  • Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
  • Safia Mahdaoui
  • Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, Tunisia
  • Taieb Ouni
  • Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, Tunisia
  • Communicated by Vicentiu D. Rădulescu.
  • Received: 2022-09-24.
  • Revised: 2022-11-04.
  • Accepted: 2022-11-05.
  • Published online: 2022-12-30.
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Cite this article as:
Sami Baraket, Safia Mahdaoui, Taieb Ouni, On the blowing up solutions of the 4-d general q-Kuramoto-Sivashinsky equation with exponentially "dominated" nonlinearity and singular weight, Opuscula Math. 43, no. 1 (2023), 5-18, https://doi.org/10.7494/OpMath.2023.43.1.5

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