Exterior scattering of non-radial solutions to energy subcritical wave equations

Abstract

We consider the defocusing, energy subcritical wave equation ${\partial }_t^{2}u-\Delta u=-|u{|}^{p-1}u$ in dimension $d\in \{3,4,5\}$ and prove the exterior scattering of solutions if $3\le d\le 5$ and $1+6/d<p<1+4/(d-2)$. More precisely, given any solution with a finite energy, there exists a solution $u_L$ to the homogeneous linear wave equation, so that the following limit holds \[ \lim_{t\rightarrow +\infty} \int_{|x|>t+R} |\nabla_{x,t} u(x,t)- \nabla_{x,t} u_L(x,t)|^2 dx = 0 \] for any fixed real number $R$. This generalize the previously known exterior scattering result in the radial case.

12 pages, 1 figure

References in corpus (1)

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