Non-simultaneous blow-up in a parabolic system with three components

• Published in: Nonlinear analysis Vol. 70; no. 5; pp. 1813 - 1829
• Main Authors: Liu, Bingchen, Li, Fengjie
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Ltd 01.03.2009; Elsevier
• Subjects: Blow-up rate; Blow-up set; Critical global existence exponent; Exact sciences and technology; Mathematical analysis; Mathematics; Non-simultaneous blow-up; Partial differential equations; Sciences and techniques of general use; Simultaneous blow-up; Simultaneous blow-up; Non-simultaneous blow-up; Blow-up set; Critical global existence exponent; Blow-up rate; primary; 35K60; Nonlinear analysis; 35B33; Parabolic equation; Blow up set; primary 35K05; 35B40; Mathematical model
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2008.02.082

This paper deals with heat equations coupled via nonlinear boundary flux ∂ u 1 ∂ η = u 1 p 11 + u 2 p 12 , ∂ u 2 ∂ η = u 2 p 22 + u 3 p 23 , ∂ u 3 ∂ η = u 3 p 33 + u 1 p 31 . A necessary and sufficient condition for the existence of only one component blowing up for nondecreasing in time and radially symmetric solutions. Three kinds of exponent regions are obtained as follows, (i) only one component blows up for every initial data; (ii) the existence of two components blowing up simultaneously while the third one remains bounded, also with two kinds of blow-up rates ( 1 2 ( p 22 − 1 ) , 1 2 ( p 33 − 1 ) ) and ( p 23 + 1 − p 33 2 ( p 33 − 1 ) , 1 2 ( p 33 − 1 ) ) ; (iii) e.g.,  u 1 remains bounded and u 2 , u 3 blow up simultaneously with blow-up rate ( p 23 + 1 − p 33 2 ( p 33 − 1 ) , 1 2 ( p 33 − 1 ) ) for every initial data. Moreover, the eight kinds of simultaneous blow-up rates and blow-up sets are discussed.