具有指数“支配”非线性和奇异权的四维广义q-Kuramoto-Sivashinsky方程的爆破解

IF 1 Q1 MATHEMATICS
S. Baraket, Safia Mahdaoui, Taieb Ouni
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引用次数: 1

摘要

设\(\Omega\)为\(\mathbb{R}^4\)中边界光滑的有界域,设\(x^{1}, x^{2}, \ldots, x^{m}\)为\(\Omega\)中的\(m\) -点。我们关注的问题是\[\Delta^{2} u - H(x,u,D^{k}u) = \rho^{4}\prod_{i=1}^{n}|x-p_{i}|^{4\alpha_{i}}f(x)g(u),\],其中主项是双拉普拉斯算子,\(H(x,u,D^{k}u)\)是一个最多对\(Du\)增长的函数,如\(|Du|^{q}\), \(1\leq q\leq 4\), \(f:\Omega\to [0,+\infty[\)是一个光滑函数,\(i = 1,\ldots, n\)满足\(f(p_{i}) \gt 0\), \(\alpha_{i}\)是正数,\(g :\mathbb R\to [0,+\infty[\)满足\(|g(u)|\leq ce^{u}\)。本文在\(\partial\Omega\)上的Navier边界条件\(u=\Delta u =0\)下,给出了\(\Omega\)上一类正弱解\((u_\rho)_{\rho\gt 0}\)存在的充分条件。当浓度集\(S=\{x^{1},\ldots,x^{m}\}\subset\Omega\)和集\(\Lambda :=\{p_{1},\ldots, p_{n}\}\subset\Omega\)不一定不相交时,当参数\( ho\)趋于0时,我们构造的解是奇异的。其证明主要基于非线性区域分解方法。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On the blowing up solutions of the 4-d general q-Kuramoto-Sivashinsky equation with exponentially "dominated" nonlinearity and singular weight
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.
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来源期刊
Opuscula Mathematica
Opuscula Mathematica MATHEMATICS-
CiteScore
1.70
自引率
20.00%
发文量
30
审稿时长
22 weeks
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