On Asymptotic Self-Similar Behaviour for a Quasilinear Heat Equation: Single Point Blow-Up

The author studies the asymptotic behaviour near a finite blow-up time $t = T$ solutions to the degenerate quasilinear parabolic equation \[ ut = (u^σ u_x )_x + u^β {in } R × (0,T),\] where σ > 0 and β > σ + 1 are fixed constants. These values of parameters σ, β correspond to single point blow-up. The initial function is assumed to be bounded, symmetric, nonincreasing in $| x |$, and compactly supported. It is proved that the resealed function f(ξ ,t) = (T - t)^1 / (β - 1) u(ξ (T - t)ᵐ ,t),m = [β - (σ + 1)] / 2(β - 1) > 0 behaves as t → T like a nontrivial self-similar profile θ (ξ ) > 0 in R,θ (ξ ) → 0 as ξ → ∞.