2024/08/29 by Qingyou He, He, Qingyou · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2408.16718
openalex publication_date 2024/08/29 · openalex created_date 2024/09/22 · openalex updated_date 2026/07/28
We consider the Cauchy problem of the porous medium type reaction-diffusion equation ∂tρ=Δρm+ρg(ρ), (x,t)∈ ℝn× ℝ+, n≥2, mgt;1, where g is the given monotonic decreasing function with the density critical threshold ρM>0 satisfying g(ρM)=0. We prove that the pressure P:=(m)/(m-1)ρm-1 in Lloc∞(ℝn) tends to the pressure critical threshold PM:=(m)/(m-1)(ρM)m-1 at the time decay rate (1+t)-1. If the initial density ρ(x,0) is compactly supported, we justify that the support \x: ρ(x,t)>0\ of the density ρ expands exponentially in time. Furthermore, we show that there exists a time T0>0 such that the pressure P is Lipschitz continuous for t>T0, which is the optimal (sharp) regularity of the pressure, and the free surface ∂ \(x,t): ρ(x,t)>0\∩ \t>T0\ is locally Lipschitz continuous. In addition, under the same initial assumptions of compact support, we verify that the free boundary ∂ \(x,t): ρ(x,t)>0\∩ \t>T0\ is a local C1,α surface.