On symmetric and nonsymmetric blowup for a weakly quasilinear heat equation

Abstract

We construct blow-up patterns for the quasilinear heat equation

$$u_t = \nabla \cdot (k(u)\nabla u) + Q(u)$$

((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 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

$$u_t = \nabla u) + u^p .$$

((SHE))

We prove that the blow-up patterns for the (QHE) and the (SHE) coincide in a structural sense under the extra assumption

$$\smallint ^\infty k(f(e^s ))ds = \infty ,$$

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.