On blow-up at space infinity for semilinear heat equations

A nonnegative blowing up solution of the semilinear heat equation ut=Δu+uput=Δu+up with p>1p>1 is considered when initial data u0u0 satisfieslim|x|→∞u0=M>0,u0⩽Mandu0≠M. It is shown that the solution blows up only at space infinity and that lim|x|→∞u(x,t)lim|x|→∞u(x,t) is the solution of the ordinary differential equation vt=vpvt=vp with v(0)=Mv(0)=M.