Large time behavior for a quasilinear diffusion equation with weighted source

Abstract

The large time behavior of general solutions to a class of quasilinear diffusion equations with a weighted source term

∂_t u = Δ uᵐ + ρ(x) uᵖ, (x,t) ∈ ℝᴺ × (0,∞),

with m > 1, 1 < p < m and suitable functions ρ(x), is established. More precisely, we consider functions ρ ∈ C(ℝᴺ) such that lim_{|x|→∞} (1+|x|)^(-σ) ρ(x) = A ∈ (0,∞), with σ ∈ (max{-N,-2},0) such that L := σ(m-1)+2(p-1) < 0. We show that, for all these choices of ρ, solutions with initial conditions u_0 ∈ C(ℝᴺ) ∩ L^∞(ℝᴺ) ∩ Lʳ(ℝᴺ) for some r ∈ [1,\infty) are global in time and, if u_0 is compactly supported, present the asymptotic behavior

lim_{t→∞} t^(-α) ||u(t) - V_*(t)||_{L^∞(ℝᴺ)} = 0,

where V_* is a suitably rescaled version of the unique compactly supported self-similar solution to the equation with the singular weight ρ(x) = |x|^σ:

U_(x,t) = tᵃ f_(|x| t^(-β)), α = -(σ+2)/L, β = -(m-p)/L.

This behavior is an interesting example of asymptotic simplification for the equation with a regular weight ρ(x) towards the singular one as t → ∞.