Weakly coupled reaction–diffusion systems with rapidly growing nonlinearities and singular initial data. (December 2019)

• Record Type: Journal Article
• Authors: Miyamoto, Yasuhito; Suzuki, Masamitsu
• Is Part Of: Nonlinear analysis. Volume 189(2019)
• Publication Date: 2019-12
• Subjects: primary 35K45 35A01 -- secondary 35B51 46E30; Existence and nonexistence -- Local in time solutions -- Weakly coupled parabolic system -- Superexponential nonlinearity; 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.2019.111576
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study existence and nonexistence of a local in time solution for the weakly coupled reaction–diffusion system ∂ t u = Δ u + g ( v ) in R N × ( 0, T ), ∂ t v = Δ v + f ( u ) in R N × ( 0, T ), ( u ( x, 0 ), v ( x, 0 ) ) = ( u 0 ( x ), v 0 ( x ) ) in R N, where f ( u ) and g ( v ) grow rapidly, u 0 and v 0 are possibly unbounded nonnegative initial functions in R N ( N ≥ 1 ) and T is a positive constant. A typical example is ( f ( u ), g ( v ) ) = ( e u p, e v q ), p ≥ 1 and q ≥ 1 . We show that if ( u 0, v 0 ) satisfies a certain integrability condition, then the local in time solution exists. Moreover, we show that there exists ( u 0, v 0 ) not satisfying the integrability condition such that the solution does not exist.