Single blowup point for a semilinear reaction‐diffusion system

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.