Local existence and nonexistence for fractional in time reaction–diffusion equations and systems with rapidly growing nonlinear terms. (September 2022)

• Record Type: Journal Article
• Authors: Suzuki, Masamitsu
• Is Part Of: Nonlinear analysis. Volume 222(2022)
• Publication Date: 2022-09
• Subjects: primary 35A01 35K15 35R11 -- secondary 26A33 46E30; Local in time solutions -- Caputo fractional derivative -- Singular initial functions -- Exponential and superexponential nonlinear terms -- Semigroup 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.2022.112909
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study the fractional in time reaction–diffusion equation ∂ t α u = Δ u + f ( u ) in R N × ( 0, T ), u ( x, 0 ) = u 0 ( x ) in R N, where 0 < α < 1, N ≥ 1, T > 0 and u 0 ≥ 0 . The fractional derivative ∂ t α is meant in a generalized Caputo sense. We mainly consider the case where f has an exponential or a superexponential growth, and u 0 has a singularity. We obtain integrability conditions on u 0 which explicitly determine local in time existence/nonexistence of a nonnegative mild solution. Moreover, our analysis can be applied to time fractional systems.