Asymptotic behavior for nonlinear degenerate parabolic equations with irregular data. Issue 16 (10th December 2021) - Niu, Weisheng; Meng, Qing; Chai, Xiaojuan
ABSTRACT: This paper focuses on the following degenerate parabolic equation 1 u t − d i v ( σ ( x ) | ∇ u | p − 2 ∇ u ) + f ( x, u ) = g i n Ω × R +, u = 0 o n ∂ Ω × R +, u ( x, 0 ) = u 0 ( x ) i n Ω, where Ω is a smooth bounded domain in R N, ( N ≥ 2 ), 1 < p < N, u 0, g ∈ L 1 ( Ω ), σ ( x ) is positive almost everywhere and satisfies proper degenerate conditions. The existence and uniqueness result is proved in the framework of entropy solutions. For the long-time behavior, we prove the existence of a global attractor in L q ( Ω ) by using some delicate estimates on the solution, which are derived by taking advantage of both the leading operator and the zero-order nonlinear term. The aforementioned results improve some previous results in the literature in several aspects.