Higher order evolution inequalities with nonlinear convolution terms. (August 2022)

• Record Type: Journal Article
• Authors: Filippucci, Roberta; Ghergu, Marius
• Is Part Of: Nonlinear analysis. Volume 221(2022)
• Publication Date: 2022-08
• Subjects: 35G20 -- 35K30 -- 35L30 -- 35B45; Higher-order evolution inequalities -- Nonlinear convolution terms -- Nonlinear capacity estimates; 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.2022.112881
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We are concerned with the study of existence and nonexistence of weak solutions to ∂ k u ∂ t k + ( − Δ ) m u ≥ ( K ∗ | u | p ) | u | q in R N × R +, ∂ i u ∂ t i ( x, 0 ) = u i ( x ) in R N, 0 ≤ i ≤ k − 1, where N, k, m ≥ 1 are positive integers, p, q > 0 and u i ∈ L loc 1 ( R N ) for 0 ≤ i ≤ k − 1 . We assume that K is a radial positive and continuous function which decreases in a neighbourhood of infinity. In the above problem, K ∗ | u | p denotes the standard convolution operation between K ( | x | ) and | u | p . We obtain necessary conditions on N, m, k, p and q such that the above problem has solutions. Our analysis emphasizes the role played by the sign of ∂ k − 1 u ∂ t k − 1 .