On symmetric and nonsymmetric blowup for a weakly quasilinear heat equation

We construct blow-up patterns for the quasilinear heat equation u_t = ∇ · (k(u)∇u) + Q(u) (QHE) in Ω × (0, T), Ω being a bounded open convex set in ℝ^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 moreover k(u) and Q(u)/u^p with a fixed p > 1 are of slow variation as u → ∞, so that (QHE) can be treated as a quasilinear perturbation of the well-known semilinear heat equation u_t = ∇u + u^p. (SHE) We prove that the blow-up patterns for the (QHE) and the (SHE) coincide in a structural sense under the extra assumption ∫^∞ k(f(e^s))ds = ∞, where f(v) is a monotone solution of the ODE f′(v) = Q(f(v))/v^p defined for all v ≫ 1. If the integral is finite then the (QHE) is shown to admit an infinite number of different blow-up patterns.