The critical exponent for semilinear σ-evolution equations with a strong non-effective damping. (February 2022)

• Record Type: Journal Article
• Authors: D'Abbicco, M.; Ebert, M.R.
• Is Part Of: Nonlinear analysis. Volume 215(2022)
• Publication Date: 2022-02
• Subjects: 35L15 -- 35L71 -- 35A01 -- 35B33 -- 35G25; Semilinear evolution equations -- Noneffective damping -- Lp−Lq estimates -- Critical exponent -- Global existence -- Small data solutions; 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.2021.112637
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we find the critical exponent for the existence of global small data solutions to: u t t + ( − Δ ) σ u + ( − Δ ) θ 2 u t = f ( u, u t ), t ≥ 0, x ∈ R n, ( u, u t ) ( 0, x ) = ( 0, u 1 ( x ) ), in the case of so-called non-effective damping, θ ∈ ( σ, 2 σ ], where σ ≠ 1 and f = | u | α or f = | u t | α, in low space dimension. By critical exponent we mean that global small data solution exists for supercritical powers α > α ̃ and do not exist, in general, for subcritical powers 1 < α < α ̃ . Assuming initial data to be small in L 1 or in some other L p space, p ∈ ( 1, 2 ), in addition to the energy space, the critical exponent only depends on the ratio n / ( σ p ) . We also prove the global existence of small data solutions in high space dimension for α > α ̄, but we leave open to determine if a counterpart nonexistence result for α < α ̄ holds or not.