2019/10/22 by Ruipeng Shen, Shen, Ruipeng
Mathematics · #35L05 #35L71 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.1910.09805
openalex publication_date 2019/10/22 · openalex created_date 2019/11/01 · openalex updated_date 2026/07/28
The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation ∂t2 u - Δu = - |u|p -1 u in the 3-dimensional space with 3≤ p<5. We generalize inward/outward energy theory and weighted Morawetz estimates for radial solutions to the non-radial case. As an application we show that if 3(5-p)/(2), then the solution scatters as long as the initial data (u0,u1) satisfy ∫_\mathbb R3 (|x|κ+1)((1)/(2)|∇ u0|2 + (1)/(2)|u1|2+(1)/(p+1)|u0|p+1) dx lt; +∞. If p=3, we can also prove the scattering result if initial data (u0,u1) are contained in the critical Sobolev space and satisfy the inequality ∫_\mathbb R3 |x|((1)/(2)|∇ u0|2 + (1)/(2)|u1|2+(1)/(4)|u0|p+1) dx lt; +∞. These assumptions on the decay rate of initial data as |x| → ∞ are weaker than previously known scattering results.