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

• Published in: Acta mathematica scientia Vol. 30; no. 6; pp. 1865 - 1880
• Main Authors: Jingxue, Yin, Liangwei, Wang, Rui, Huang
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01-11-2010; Department of Mathematics, Jilin University, Changchun 130012, China%School of Mathematical Sciences, South China Normal University, Guangzhou 510031, China
• Subjects: 35B40; 35K65; asymptotic behavior; Asymptotic properties; Cauchy problem; Complexity; Mathematical analysis; porous medium equation; complexity; porous medium equation; asymptotic behavior; 35B40; 35K65
• ISSN: 0252-9602; 1572-9087
• DOI: 10.1016/S0252-9602(10)60179-8

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.