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

✍ Scribed by Bingchen Liu; Fengjie Li

• Publisher: Elsevier Science
• Year: 2009
• Tongue: English
• Weight: 981 KB
• Volume: 70
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2008.02.082

✦ Synopsis

This paper deals with heat equations coupled via nonlinear boundary flux ∂u 1 ∂η = u p 11 1 + u p 12 2 , ∂u 2 ∂η = u p 22 2 + u p 23 3 , ∂u 3 ∂η = u p 33 3 + u p 31

1 . 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.