2015/01/01 by Ruipeng Shen, Shen, Ruipeng
Mathematics · #35L71 35L05 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35L05 #msc:35L71
paper · pdf · doi:10.48550/arxiv.1501.00323
arxiv created 2015/01/01 · openalex publication_date 2015/01/01 · arxiv updated 2015/01/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work we consider a semi-linear energy critical wave equation in \mathbb Rd (3≤ d ≤ 5) ∂t2 u - Δu = ± ϕ(x) |u|4/(d-2) u, (x,t)∈ \mathbb Rd × \mathbb R with initial data (u, ∂t u)|t=0 = (u0,u1) ∈ H1 × L2 (\mathbb Rd). Here the function ϕ∈ C(\mathbb Rd; (0,1]) converges to zero as |x| → ∞. We follow the same compactness-rigidity argument as Kenig and Merle applied on the Cauchy problem of the equation ∂t2 u - Δu = |u|4/(d-2) u and obtain a similar result when ϕ satisfies some technical conditions. In the defocusing case we prove that the solution scatters for any initial data in the energy space H1 × L2. While in the focusing case we can determine the global behaviour of the solutions, either scattering or finite-time blow-up, according to their initial data when the energy is smaller than a certain threshold.