Blow-up profiles in one-dimensional, semilinear parabolic problems

Abstract

Let u be a solution of the Cauchy problem ut=uxx+up, x∈R, t>0, u(x,0)=u0(x), x∈R, where p>1 and u0 is continuous, nonnegative, and bounded. Suppose that u blows up at t=T<∞ and u(x,t)≢(p−1)−1/(p−1)(T−t)−1/(p−1). The authors show that the blow-up set is discrete. Also, if x=0 is a blow-up point then either limx→0[|x|2/log|x|]1/(p−1)u(x,T)=[8p/(p−1)2] 1/(p−1) or there exists a constant C>0 and an even integer m≥4 such that limx→0|x|m/(p−1)u(x,T)=C.