A test function method for evolution equations with fractional powers of the Laplace operator. (January 2021)

• Record Type: Journal Article
• Authors: D'Abbicco, M.; Fujiwara, K.
• Is Part Of: Nonlinear analysis. Volume 202(2021)
• Publication Date: 2021-01
• Subjects: Nonlinear evolution equations -- Critical exponent -- Global-in-time solutions -- Fractional Laplacian -- Test function method; 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.2020.112114
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we discuss a test function method to obtain nonexistence of global-in-time solutions for higher order evolution equations with fractional derivatives and a power nonlinearity, under a sign condition on the initial data. In order to deal with fractional powers of the Laplace operator, we introduce a suitable test function and a suitable class of weak solutions. The optimality of the nonexistence result provided is guaranteed by both scaling arguments and counterexamples. In particular, our manuscript provides the counterpart of nonexistence for several recent results of global-in-time existence of small data solutions to the following problem: u t t + ( − Δ ) θ u t + ( − Δ ) σ u = f ( u, u t ), t > 0, x ∈ R n, u ( 0, x ) = u 0 ( x ), u t ( 0, x ) = u 1 ( x ), with f = | u | p or f = | u t | p, where θ ≥ 0 and σ > 0 are fractional powers.