Inward/outward energy theory of non-radial solutions to wave equations

Work of Ruipeng Shen

In this work we consider an energy sub-critical, defocusing wave equation with a power nonlinearity in the 3-dimensional space. We split the total energy into a sum of inward energy and outward energy, then discuss the properties of energy distribution and transformation. This gives plentiful information about the asymptotic behaviour of finite-energy solutions. More precisely, we prove that almost all energy is inward energy as time approaches negative infinity. Then inward energy gradually transforms into outward energy. Finally almost all energy becomes outward energy as time approach positive infinity.

In addition, if the initial data decay at a certain rate as the spatial variable tends to infinity, then the inward energy decays at a similar rate as time tends to infinity. As an application, we may combine this with a suitable local theory to prove the scattering of solutions if the initial data decay sufficiently fast. Our assumption on the decay rate is weaker than previously known results.

Adv. Math.

https://doi.org/10.1016/j.aim.2020.107384