2017/11/06 by Xiangsheng Xu, Xu, Xiangsheng
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1711.01965
openalex publication_date 2017/11/06 · openalex created_date 2017/11/17 · openalex updated_date 2026/07/28
It is well known that a weak solution φ to the initial boundary value problem for the uniformly parabolic equation ∂tφ-div(A∇ φ) +ωφ= f in ΩT≡Ω×(0,T) satisfies the uniform estimate ‖φ‖∞,ΩT≤ ‖φ‖∞,∂pΩT+c‖f‖q,ΩT, c=c(N,λ, q, ΩT), provided that q>1+(N)/(2), where Ω is a bounded domain in ℝN with Lipschitz boundary, T>0, ∂pΩT is the parabolic boundary of ΩT, ω∈ L1(ΩT) with ω≥ 0, and λ is the smallest eigenvalue of the coefficient matrix A. This estimate is sharp in the sense that it generally fails if q=1+(N)/(2). In this paper we show that the linear growth of this upper bound in ‖f‖q,ΩT can be improved. To be precise, we establish ‖φ‖∞,ΩT≤ ‖φ0‖∞,∂pΩT+c‖f‖1+(N)/(2),ΩT(ln(‖f‖q,ΩT+1)+1).