Weakly non-radiative radial solutions to 3D energy subcritical wave equations

2020/05/24 by Ruipeng Shen, Shen, Ruipeng
Mathematics · #35L05 #35L71 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35L05 #msc:35L71

paper · pdf · doi:10.48550/arxiv.2005.11681

22 pages, 2 figures

arxiv created 2020/05/24 · arxiv updated 2020/05/26

Abstract

In this work we consider the energy subcritical 3D wave equation ∂_t² u - Δu = ± |u|^p-1 u and discuss its (weakly) non-radiative solutions, i.e. the solutions defined in an exterior region \(x,t): |x|>|t|+R\ with R≥ 0 satisfying lim_{t→ ±∞} ∫_|x|>|t|+R (|∇ u(x,t)|² + |u_t(x,t)|²) dx = 0. It has been known that any radial weakly non-radiative solution to the linear wave equation is a multiple of 1/|x|. In addition, any radial weakly non-radiative solutions u to the energy critical wave equation must possess a similar asymptotic behaviour, i.e. u(x,t)≃ C/|x| when |x| is large. In this work we give examples to show that radial weakly non-radiative solutions to energy subcritical equation (3<p<5) may possess a much different asymptotic behaviour. However, a radial weakly non-radiative solution u with initial data in the critical Sobolev space H^s_p× H^s_p-1(ℝ³) must coincide with a C² solution W to the elliptic equation -ΔW = -|W|^p-1 W so that u(x,t) ≡ W(x) ≃ C/|x| when |x| is large.

Related

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• On classification of non-radiative solutions for various energy-critical wave equations
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• Asymptotic behaviour of non-radiative solution to the wave equations