Energy Distribution of Radial Solutions to Energy Subcritical Wave Equation with an Application on Scattering Theory

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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 (3≤ p<5) whose initial data are radial and come with a finite energy. We split the energy into inward and outward energies, then apply energy flux formula to obtain the following asymptotic distribution of energy: Unless the solution scatters, its energy can be divided into two parts: "scattering energy" which concentrates around the light cone |x|=|t| and moves to infinity at the light speed and "retarded energy" which is at a distance of at least |t|β behind when |t| is large. Here β is an arbitrary constant smaller than β₀(p) = 2(p-2)/p+1. A combination of this property with a more detailed version of the classic Morawetz estimate gives a scattering result under a weaker assumption on initial data (u₀,u₁) than previously known results. More precisely, we assume ∫R³ (|x|κ+1)(1/2|∇ u₀|² + 1/2|u₁|²+1/p+1|u|ᵖ⁺¹) dx < +∞. Here κ>κ₀(p) =1-β₀(p) = 5-p/p+1 is a constant.