On symmetric and nonsymmetric blowup for a weakly quasilinear heat equation

We construct blow-up patterns for the quasilinear heat equation (QHE) $$u_t = nabla cdot (k(u)nabla u) + Q(u)$$ in Ω×(0,T), Ω being a bounded open convex set in ? ^N with smooth boundary, with zero Dirichet boundary condition and nonnegative initial data. The nonlinear coefficients of the equation are assumed to be smooth and positive functions and moreoverk(u) andQ(u)/u ^p with a fixedp>1 are of slow variation asu→∞, so that (QHE) can be treated as a quasilinear perturbation of the well-known semilinear heat equation (SHE) $$u_t = nabla u) + u^p .$$ We prove that the blow-up patterns for the (QHE) and the (SHE) coincide in a structural sense under the extra assumption $$smallint ^infty k(f(e^s ))ds = infty ,$$ wheref(v) is a monotone solution of the ODEf′(v)=Q(f(v))/v ^p defined for allv?1. If the integral is finite then the (QHE) is shown to admit an infinite number of different blow-up patterns.