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- W2888403921 abstract "The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation $partial_t^2 u - Delta u = - |u|^{p -1} u$ in the 3-dimensional space ($3leq p<5$) 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 $|t|^beta$ behind when $|t|$ is large. Here $beta$ is an arbitrary constant smaller than $beta_0(p) = frac{2(p-2)}{p+1}$. 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 $(u_0,u_1)$ than previously known results. More precisely, we assume [ int_{{mathbb R}^3} (|x|^kappa+1)left(frac{1}{2}|nabla u_0|^2 + frac{1}{2}|u_1|^2+frac{1}{p+1}|u|^{p+1}right) dx < +infty. ] Here $kappa>kappa_0(p) =1-beta_0(p) = frac{5-p}{p+1}$ is a constant." @default.
- W2888403921 created "2018-08-31" @default.
- W2888403921 creator A5084151289 @default.
- W2888403921 date "2018-08-26" @default.
- W2888403921 modified "2023-10-16" @default.
- W2888403921 title "Energy Distribution of Radial Solutions to Energy Subcritical Wave Equation with an Application on Scattering Theory" @default.
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- W2888403921 doi "https://doi.org/10.48550/arxiv.1808.08656" @default.
- W2888403921 hasPublicationYear "2018" @default.
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