Explosion de solutions d'équations paraboliques semilinéaires supercritiques

Abstract

The authors consider blow-up for the equation (1) ut=Δu+up (x∈RN, t>0), where p>1 and N>1. For N>11and (2) p>(N−2(N−1)1/2)/(N−4−2(N−1)1/2)=p1(N) there exist some radial positive solutions that blow up at x=0, t=T<∞. Moreover, (3) limsup(T−t)1/(p−1)u(0,t)=∞ (t→T). Similar problems were investigated in detail in the book by A. A. Samarskiĭ et al. [Peaking modes in problems for quasilinear parabolic equations (Russian), "Nauka'', Moscow, 1987] and in other works where blow-up was established under conditions of the type 1<p<p2(N) with p2<p1. For corresponding solutions the lim sup in (3) is bounded. The authors give some arguments which show the following. The true threshold p that separates solutions with bounded and unbounded limit (3) should have the form p=p1(N).