Energy distribution of solutions to defocusing semi-linear wave equation in higher dimensional space

Abstract

The topic of this paper is a semi-linear, defocusing wave equation $u_{tt}-\Delta u=-|u{|}^{p-1}u$ in sub-conformal case in the higher dimensional space whose initial data are radical and come with a finite energy. We prove some decay estimates of the the solutions if initial data decay at a certain rate as the spatial variable tends to infinity. A combination of this property with a method of characteristic lines give a scattering result if the initial data satisfy $$E_{\kappa }\left(u_0,u_1\right)={\int }_{{\mathbb{R}}^d}\left(|x{|}^{\kappa }+1\right)\left(\frac{1}{2}{\left|\nabla u_0(x)\right|}^2+\frac{1}{2}{\left|u_1(x)\right|}^2+\frac{1}{p+1}{\left|u_0(x)\right|}^{p+1}\right)dx<+\infty .$$ Here $\kappa =\frac{(2-d)p+(d+2)}{p+1}$.

15 pages. arXiv admin note: text overlap with arXiv:2104.13041

References in corpus (1)

• Inward/outward Energy Theory of Wave Equation in Higher Dimensions