Weighted Lp estimates and Fujita exponent for a nonlocal equation. (July 2019)

• Record Type: Journal Article
• Authors: Khomrutai, Sujin
• Is Part Of: Nonlinear analysis. Volume 184(2019)
• Page Start: 321
• Page End: 351
• Publication Date: 2019-07
• Subjects: 35A01 -- 35B40 -- 35B33 -- 35B44 -- 35K70 -- 45M05 -- 45G05; Weighted Lp estimates -- Nonlocal equations -- Pseudoparabolic equations -- Blow-up solutions -- Global solutions -- Fujita exponent -- Unbounded coefficient; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.02.027
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We investigate a nonlocal equation ∂ t u = ∫ R n J ( x − y ) u ( y, t ) d y − ‖ J ‖ L 1 u ( x, t ) + a ( x, t ) u p in R n, where a is unbounded and J belongs to a weighted space. Crucial weighted L p and interpolation estimates for the Green operator are established by a new method based on the sharp Young's inequality, the asymptotic behavior of a regular varying coefficients exponential series, and the properties of auxiliary functions Γ = ( 1 + | x | 2 ∕ η ) b ∕ 2 that − Γ ∕ η ≲ J ∗ Γ − Γ ≲ Γ ∕ η and η − b + ∕ 2 ≲ Γ ∕ x b ≲ η − b − ∕ 2 . Blow-up behaviors are investigated by employing test functions ϕ R = Γ ( η = R ) instead of principal eigenfunctions. Global well-posedness in weighted L p spaces for the Cauchy problem is proved. When a ∼ x σ the Fujita exponent is shown to be 1 + ( σ + 2 ) ∕ n . Our approach generalizes and unifies nonlocal diffusion equations and pseudoparabolic equations.