Existence and asymptotic behaviour of solutions of the very fast diffusion equation

Abstract

Let n ≥ 3, 0 < m ≤ (n − 2)/n, p > max(1, (1 − m)n/2), and \({0 \le u_0 \in L_{loc}^p(\mathbb{R}^n)}\) satisfy \({{\rm lim \, inf}_{R\to\infty}R^{-n+\frac{2}{1-m}} \int_{|x|\le R}u_0\,dx = \infty}\). We prove the existence of unique global classical solution of u t = Δu m, u > 0, in \({\mathbb{R}^n \times (0, \infty), u(x, 0) = u_0(x)}\) in \({\mathbb{R}^n}\). If in addition 0 < m < (n − 2)/n and u 0(x) ≈ A|x|−q as |x| → ∞ for some constants A > 0, q < n/p, we prove that there exist constants α, β, such that the function v(x, t) = t α u(t β x, t) converges uniformly on every compact subset of \({\mathbb{R}^n}\) to the self-similar solution ψ(x, 1) of the equation with ψ(x, 0) = A|x|−q as t → ∞. Note that when m = (n − 2)/(n + 2), n ≥ 3, if \({g_{ij} = u^{\frac{4}{n+2}}\delta_{ij}}\) is a metric on \({\mathbb{R}^n}\) that evolves by the Yamabe flow ∂g ij /∂t = −Rg ij with u(x, 0) = u 0(x) in \({\mathbb{R}^n}\) where R is the scalar curvature, then u(x, t) is a global solution of the above fast diffusion equation.