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Single blowup point for a semilinear reaction‐diffusion system
Author(s) -
Zhang Zhengce,
Huang Yaodan
Publication year - 2018
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201400335
Subject(s) - semigroup , mathematics , reaction–diffusion system , point (geometry) , diffusion , set (abstract data type) , set point , pure mathematics , mathematical analysis , geometry , computer science , thermodynamics , physics , control engineering , engineering , programming language
Abstract Throughout this paper, we investigate the blowup set for the semilinear reaction‐diffusion systemu t=Δ u + f ( u , v ) ,x ∈ Ω , t > 0 ,v t=Δ v + g ( u , v ) ,x ∈ Ω , t > 0 ,u ( x , t )=v ( x , t ) = 0 ,x ∈ ∂ Ω , t > 0 ,u ( x , 0 )=u 0 ( x ) ,v ( x , 0 ) = v 0 ( x ) ,x ∈ Ω ,where Ω = B R : = { x ∈ R n ; | x | < R } andu 0 ( x ) , v 0 ( x ) ∈ L ∞ ( Ω ) . The initial datau 0 ( x )andv 0 ( x )are positive, radially symmetric and decreasing. Under certain assumptions on f and g , we prove that the solution of this system blows up only at the origin. The proof is based on the Friedman–McLeod method, comparison principle and semigroup method.