Complexity of asymptotic behavior of solutions for the porous medium equation with absorption

Abstract

In this paper we analyze the large time behavior of nonnegative solutions of the Cauchy problem of the porous medium equation with absorption ut−Δum+γup=0, where γ ≥ 0, m > 1 and p>m>2N. We will show that if γ = 0 and 0<μ<2NN(m−1)+2, or γ > 0 and 1p−1<μ<2NN(m−1)+2, then for any nonnegative function ϕ in a nonnegative countable subset F of the Schwartz space S(ℝN), there exists an initial-value u0∈C(ℝN) with limx→∞u0(x)=0 such that ϕ is an Ω-limit point of the rescaled solutions tμ2u(tβ•,t), where β=2−μ(m−1)4.