On the existence with exponential decay and the blow-up of solutions for coupled systems of semi-linear corner-degenerate parabolic equations with singular potentials

• Published in: Acta mathematica scientia Vol. 41; no. 1; pp. 257 - 282
• Main Authors: Chen, Hua, Liu, Nian
• Format: Journal Article
• Language: English
• Published: Singapore Springer Singapore 01.01.2021; School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China%School of Science, Wuhan University of Technology, Wuhan 430070, China
• Subjects: Analysis; Mathematics; Mathematics and Statistics; totally characteristic degeneracy; 35B35; blow-up; 35B44; coupled parabolic equations; asymptotic stability; singular potentials; 35K65; singular poten-tials
• ISSN: 0252-9602; 1572-9087
• DOI: 10.1007/s10473-021-0115-3

In this article, we study the initial boundary value problem of coupled semi-linear degenerate parabolic equations with a singular potential term on manifolds with corner singularities. Firstly, we introduce the corner type weighted p -Sobolev spaces and the weighted corner type Sobolev inequality, the Poincaré inequality, and the Hardy inequality. Then, by using the potential well method and the inequality mentioned above, we obtain an existence theorem of global solutions with exponential decay and show the blow-up in finite time of solutions for both cases with low initial energy and critical initial energy. Significantly, the relation between the above two phenomena is derived as a sharp condition. Moreover, we show that the global existence also holds for the case of a potential well family.