Composite structure of global unbounded solutions of nonlinear heat equations with critical Sobolev exponents

Abstract

We describe special asymptotic structures of solutions of the semilinear heat equation u t= Δu+u p in Ω×(0,T), u=0 on ∂Ω×(0,T), in the unit ball Ω={r=|x|<1}⊂ R N in dimensions N⩾3 with positive symmetric initial data u 0( r). It is known (J. Differential Equations 54 (1984) 97) that in the critical Sobolev case p= p s =( N+2)/( N−2) the Cauchy problem with specially chosen initial data u 0 admits global unbounded solutions (GUSs) u(·, t), which are uniformly bounded in L 1(Ω) but are not bounded in L ∞(Ω) for all t>0. In the radial geometry, we establish the following asymptotic behaviour of such solutions as t→∞: ln ||u(·,t)|| ∞∼π 2t/4 for N=3, ln||u(·,t)|| ∞∼2 t for N=4, and || u(·, t)|| ∞∼ γ 0 t ( N−2)/2( N−4) for N⩾5, where γ 0= γ 0( N)>0 is a constant independent of initial data. The large-time behaviour is not self-similar and is obtained by matching of inner and outer asymptotic expansions. The phenomenon of GUSs is shown to be a common feature of a number of quasilinear and fully nonlinear parabolic equations with scaling invariant operators in the critical Sobolev case.