Nonexistence of backward self-similar blowup solutions to a supercritical semilinear heat equation

We consider a Cauchy problem for a semilinear heat equation{ut=Δu+upin RN×(0,T),u(x,0)=u0(x)⩾0in RN with p>pSp>pS where pSpS is the Sobolev exponent. If u(x,t)=(T−t)−1/(p−1)φ((T−t)−1/2x)u(x,t)=(T−t)−1/(p−1)φ((T−t)−1/2x) for x∈RNx∈RN and t∈[0,T)t∈[0,T), where φ is a regular positive solution ofequation(P)Δφ−y2∇φ−1p−1φ+φp=0in RN, then u   is called a backward self-similar blowup solution. It is immediate that (P) has a trivial positive solution κ≡(p−1)−1/(p−1)κ≡(p−1)−1/(p−1) for all p>1p>1. Let pLpL be the Lepin exponent. Lepin obtained a radial regular positive solution of (P) except κ   for pSpLp>pL.