Local well-posedness and small data scattering for energy super-critical nonlinear wave equations. Issue 3 (17th February 2021) - Gao, Yili; Xue, Jun
Abstract : In this work, we consider the following nonlinear wave equations ∂ t t u − Δ u + | u | p u = 0, ( t, x ) ∈ R × R N . We prove that when p > 4 N − 2 and 3 ≤ N ≤ 9 ; or N ≥ 10, p < N 2 − 4 N + 1 − N 4 − 8 N 3 − 14 N 2 + 56 N − 31 4 ( N − 1 ) . The Cauchy problem is locally well-posed in H ˙ s c ( R N ) × a ˙ H s c − 1 ( R N ) with s c = N 2 − 2 p . Moreover, the small data theory holds under the same restriction.