Scattering of solutions to the defocusing energy sub-critical semi-linear wave equation in 3D

الأصل · EN

In this paper we consider a semi-linear, energy sub-critical, defocusing wave equation ∂ₜ² u - Δu = - |u|ᵖ ⁻¹ u in the 3-dimensional space with p ∈ [3,5). We prove that if initial data (u₀, u₁) are radial so that ∇ u₀L² (R³; dμ), u₁L² (R³; dμ) ≤ ∞, where d μ= (|x|+1)¹⁺²ε with ε > 0, then the corresponding solution u must exist for all time t ∈ R and scatter. The key ingredients of the proof include a transformation T so that v = T u solves the equation vττ - Δy v = - (|y|/ |y|)ᵖ⁻¹ e⁻⁽ᵖ⁻³⁾τ |v|ᵖ⁻¹v with a finite energy, and a couple of global space-time integral estimates regarding a solution v as above.