Complexity of asymptotic behavior of the porous medium equation in ${\mathbb{R}^N}$

Abstract

In this paper, we consider the complexity of large time behavior of solutions to the porous medium equation u _t − Δu ^m = 0 in ${\mathbb{R}^N}$ with m > 1. We first show that for any given ${0<\mu<\frac{2N}{N(m-1)+2}}$ and ${\beta>\frac{2-\mu(m-1)}{4}}$, the ω-limit set of ${t^{\frac{\mu}{2}}u(t^{\beta}\cdot,t)}$ includes all of the nonnegative functions f in the Schwartz space ${\fancyscript{S}(\mathbb{R}^N)}$ with f(0) = 0. Furthermore, we prove that, for a given countable subset E of the interval ${\left(0,\frac{2N}{(N(m-1)+2)(2+\mu(m-1))}\right)}$, there exists an initial value u ₀(x) such that for all μ and β satisfying ${0<\mu<\frac{2N}{N(m-1)+2}, \beta></\frac{2N}{N(m-1)+2},>\frac{2-\mu(m-1)}{4}}$ and ${\frac{\mu}{2\beta}\in E}$, the ω-limit set of ${t^{\frac{\mu}{2}}u(t^{\beta}\cdot,t)}$ is equal to ${C_{0}^{+}(\mathbb{R}^N)\equiv\{f\in C_{0}(\mathbb{R}^N);\ f(x)\geq 0,\ f(0)=0\}}$.