Nonlinear Degenerate Parabolic Equations with Time-dependent Singular Potentials for Baouendi-Grushin Vector Fields

• Published in: Acta mathematica Sinica. English series Vol. 31; no. 1; pp. 123 - 139
• Main Authors: Han, Jun Qiang, Guo, Qian Qiao
• Format: Journal Article
• Language: English
• Published: Heidelberg Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 2015; Springer Nature B.V
• Subjects: Boundaries; Cylinders; Euclidean space; Fields (mathematics); Inequality; Mathematical analysis; Mathematics; Mathematics and Statistics; Nonlinear equations; Nonlinearity; Studies; Texts; 不存在性; 向量; 奇异; 时间依赖性; 时间相关; 退化抛物方程; 退缩抛物方程; 非线性; positive solutions; nonexistence; Hardy inequality; Baouendi-Grushin vector fields; Nonlinear degenerate parabolic equations; 35K55; 35K65
• ISSN: 1439-8516; 1439-7617
• DOI: 10.1007/s10114-015-3757-z

In this paper, we are concerned with the following three types of nonlinear degenerate parabolic equations with time-dependent singular potentials： uq/ t=▽α·（‖z‖^-pγ｜▽αu｜^p-2▽αu）＋V（z, t）u^p-1, uq/ t=▽α·（‖z‖^-2γ▽αu^m）＋V（z, t）u^m, uq/ t=u^μ▽α·（u^τ｜▽αu｜^p-2▽αu）＋V（z, t）u^p-1＋μ＋τ in a cylinder Ω×（0, T） with initial condition u（z, 0）=u0（z） ≥ 0 and vanishing on the boundary Ω×（0, T）, where Ω is a Carnot-Carathéodory metric ball in Rd＋k and the time-dependent singular potential function is V（z, t） ∈ L^1loc （Ω×（0, T））. We investigate the nonexistence of positive solutions of these three problems and present our results on nonexistence.