Large time behavior of ODE type solutions to nonlinear diffusion equations

2018/09/12 by Junyong Eom, Kazuhiro Ishige, Eom, Junyong +1
Computer Science · Mathematics · #35B40 #35K55 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1809.04252

openalex publication_date 2018/09/12 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

Consider the Cauchy problem for a nonlinear diffusion equation \ ∂_t u=Δu^m+u^α · amp; \quadin \bf R^N×(0,∞),
u(x,0)=λ+φ(x) · gt;0 · amp; \quadin \bf R^N, . where m>0, α∈(-∞,1), λ>0 and φ∈ BC(\bf R^N) ∩ L^r(\bf R^N) with 1≤ r-λ. Then the positive solution to problem (P) behaves like a positive solution to ODE ζ'=ζ^α in (0,∞) and it tends to +∞ as t→∞. In this paper we obtain the precise description of the large time behavior of the solution and reveal the relationship between the behavior of the solution and the diffusion effect the nonlinear diffusion equation has.

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