2024/05/01 by Ikki Fukuda, Fukuda, Ikki, Shinya Sato +1
Computer Science · Mathematics · #35B40 #35K15 #35K55 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2405.00896
openalex publication_date 2024/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the asymptotic behavior of solutions to the convection-diffusion equation: ∂t u - div(a(x)∇ u) = d⋅∇ (| u| q-1u), x∈ℝn, tgt;0 with an integrable initial data u0(x), where n≥1, q>1+(1)/(n) and d∈ ℝn. Moreover, we take a(x)=1+b(x)>0, where b(x) is smooth and decays fast enough at spatial infinity. It is known that the asymptotic profile of the solution to this problem can be given by the heat kernel. Moreover, some higher-order asymptotic expansions of the solution have already been studied. In particular, the structures of the second asymptotic profiles strongly depend on the nonlinear exponent q. More precisely, these profiles have different decay orders in each of the following three cases: 1+(1)/(n)1+(2)/(n). In this paper, we focus on the critical case q=1+(2)/(n). By analyzing the corresponding integral equation in details, we have succeeded to give the more higher-order asymptotic expansion of the solution, which generalizes the previous works.