A comparison principle for nonlinear parabolic equations with nonlocal source and gradient absorption

2025/05/13 by Amirzhankyzy, Zhaniya, Yessirkegenov, Nurgissa
#35B09 #35B51 #35K55 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.08753

Abstract

This paper investigates the initial-boundary value problem for a nonlinear parabolic equation involving the p-Laplacian operator, nonlocal source terms, gradient absorption, and various nonlinearities: (∂ u)/(∂ t) - div(|∇ u|^p-2 ∇ u ) = α|u|^k-1u ∫_Ω|u|^s dx - β|u|^l-1u |∇ u|^q + γu^m + μ|∇ u|^r - ν|u|^σ-1u, where Ω is a bounded domain in ℝ^N, N ≥ 1, with a smooth boundary ∂ Ω. The parameters satisfy α, l, σ> 0 , β, ν≥ 0 , k, m, s ≥ 1 , r ≥ p - 1 ≥ (p)/(2), and γ, μ∈ ℝ. We establish a comparison principle for this problem. Using this principle, we derive blow-up results as well as global-in-time boundedness of solutions. Our results extend and unify previous studies in the literature.

Citations

• A comparison principle for higher order nonlinear hypoelliptic heat operators on graded Lie groups
• Gradient estimates for a degenerate parabolic equation with gradient absorption and applications

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