Attractors for a class of semi-linear degenerate parabolic equations with critical exponent

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• 作者：Dandan Li ; Chunyou Sun
• 关键词：Mathematics Subject Classification35L70 ; 35L90 ; 37L30
• 刊名：Journal of Evolution Equations
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：16
• 期：4
• 页码：997-1015
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis;
• 出版者：Springer International Publishing
• ISSN：1424-3202
• 卷排序：16

文摘

We consider the degenerate parabolic equation \({ \partial_t u = \triangle_{\lambda}u + f(u)}\) with Dirichlet boundary condition defined on a bounded domain \({\Omega \subset \mathbb{R}^N}\), where \({\triangle_{\lambda}}\), the so-called \({\triangle_{\lambda}}\) -Laplacian, is a subelliptic operator of the type$$\triangle_{\lambda} :=\sum_{i=1}^N\partial_{x_i}(\lambda^2_i\partial_{x_i}),\quad \lambda= (\lambda_1(x),\ldots,\lambda_N(x)).$$In this paper, we will establish the global existence of solutions and its corresponding attractor with critical nonlinearity$$|f(u)-f(v)| \leq c|u-v|\left(1+|u|^{\frac{4}{Q-2}}+|v|^{\frac{4}{Q-2}}\right),\quad\forall~~u,v\in \mathbb{R}.$$Mathematics Subject Classification35L7035L9037L30This work was partly supported by the NSFC (Grants No. 11471148, 11522109).