Single blowup point for a semilinear reaction‐diffusion system

Abstract

Throughout this paper, we investigate the blowup set for the semilinear reaction‐diffusion system rightut=leftnormalΔu+ffalse(u,vfalse),leftx∈normalΩ,0.33emt>0,rightvt=leftnormalΔv+gfalse(u,vfalse),leftx∈normalΩ,0.33emt>0,rightufalse(x,tfalse)=leftvfalse(x,tfalse)=0,leftx∈∂normalΩ,0.33emt>0,rightufalse(x,0false)=leftu0(x),0.33em0.33emv(x,0)=v0(x),leftx∈normalΩ,where Ω=BR:={x∈double-struckRn;false|xfalse|<R} and u0false(xfalse),v0false(xfalse)∈L∞false(normalΩfalse). The initial data u0false(xfalse) and v0false(xfalse) 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.