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 − u m = 0 in R N with m > 1. We first show that for any given 0Furthermore, we prove that, for a given countable subset E of the interval 0, 2N (N (m−1)+2)(2+μ(m−1)) , there exists an initial value u 0 (x) such that for all μ and β

$\omega$-limit sets for porous medium equation with initial data in some weighted spaces

“…for ϕ ∈ L 1 loc (R N ). Just as did in [20], from the definitions of S(t) and D σ λ , we can get the following commutation rule…”

Remark on the Cauchy problem for the evolution p-Laplacian equation

“…In 2002, it was Vázquez and Zuazua [ 6 ] who first successfully used the ω -limit set of the rescaled solutions to study the complicated asymptotic behavior of solutions for the problem ( 1.1 )-( 1.2 ). Subsequently, we [ 7 – 10 ] investigated the complicated asymptotic behavior of solutions of the porous medium equation by using the ω -limit set of the rescaled solutions with . Using ω -limit set to research other partial differential equations, one can refer to [ 11 – 14 ].…”