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

Abstract

We consider a Cauchy problem for a semilinear heat equation { u t = Δ u + u p in R N × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) ⩾ 0 in R N with p > p S where p S is the Sobolev exponent. If u ( x , t ) = ( T − t ) − 1 / ( p − 1 ) φ ( ( T − t ) − 1 / 2 x ) for x ∈ R N and t ∈ [ 0 , T ) , where φ is a regular positive solution of (P) Δ φ − y 2 ∇ φ − 1 p − 1 φ + φ p = 0 in R N , 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 ) for all p > 1 . Let p L be the Lepin exponent. Lepin obtained a radial regular positive solution of (P) except κ for p S < p < p L . We show that there exist no radial regular positive solutions of (P) which are spatially inhomogeneous for p > p L .