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

We describe special asymptotic structures of solutions of the semilinear heat equation ut=Δu+up in Ω×(0,T), u=0 on ∂Ω×(0,T),in the unit ball Ω={r=x<1} RN in dimensions N 3 with positive symmetric initial data u0(r). It is known (J. Differential Equations 54 (1984) 97) that in the critical Sobolev case p=ps=(N+2)/(N−2) the Cauchy problem with specially chosen initial data u0 admits global unbounded solutions (GUSs) u(•,t), which are uniformly bounded in L1(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→∞: and u(•,t)∞ γ0t(N−2)/2(N−4) for N 5, where γ0=γ0(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.