On the uniqueness of L1L1-continuation after blowup

This paper is concerned with the uniqueness of L1L1-continuation beyond blowup for a Cauchy problem of a semilinear heat equationequation(P){ut=Δu+upin RN×(0,T˜),u(x,0)=u0(x)⩾0in RN with p>1p>1, 0pJLp>pJL in the radial case. If for an L1L1-solution u   of (P) there exists a sequence {un}{un} of classical solutions of (P) such that u0,n→u0u0,n→u0 in L∞(RN)L∞(RN) as n→∞n→∞ for the sequence {u0,n}{u0,n} of initial data and that un(t)→u(t)un(t)→u(t) in Llocp(RN) as n→∞n→∞ for t∈(0,T˜), then u   is called a limit L1L1-solution. Based on the sufficient condition, we prove the uniqueness of limit L1L1-solution with radial symmetry after blowup for p>pJLp>pJL.