Scattering theory for the defocusing energy-supercritical nonlinear wave equation with a convolution. (March 2017) - Han, Wei
Abstract: In this paper, we study the global well-posedness and scattering problem for the nonlinear wave equation with a convolution u t t − Δ u + ( | x | − γ ∗ | u | 2 ) u = 0 in dimensions d ≥ 6 . We show that if the solution u is apriorily bounded in the critical homogeneous Sobolev space, that is, u ∈ L t ∞ ( H ̇ x s c × H ̇ x s c − 1 ), with s c ≔ γ − 2 2 > 1, then u is global and it scatters. The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation and nonlinear Schrödinger equation. Our analysis derived from the concentration compactness method to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of three scenarios: the finite time blow-up solution, the soliton-like solution and the low-to-high frequency cascade. We note that, authors preclude the finite time blow-up solution to wave equation usually by the property of the finite speed of propagation in the previous literature, however, the finite speed of propagation is broken in the nonlocal nonlinear wave equation with a convolution nonlinearity. For this, we will initially establish the low regularity results for almost periodic solutions including the finite time blow-up solutions and initially preclude the finite time blow-up solutions by lowering the regularity.