We consider nonlinear half-wave equations with focusing power-type nonlinearity i∂tu = p−∆ u − |u|p−1u, with (t, x) ∈ R × Rd with exponents 1 < p < ∞ for d = 1 and 1 < p < (d + 1)/(d − 1) for d ≥ 2. We study traveling solitary waves of the form u(t, x) = eiωtQv(x − vt) with frequency ω ∈ R, velocity v ∈ Rd, and some finite-energy profile Qv ∈ H1/2(Rd), Qv 6≡ 0. We prove that traveling solitary waves for speeds |v| ≥ 1 do not exist. Furthermore, we generalize the non-existence result to the square root Klein-Gordon operator √−∆ + m2 and other nonlinearities. As a second main result, we show that small data scattering fails to hold for the focusing half-wave equation in any space dimension. The proof is based on the existence and properties of traveling solitary waves for speeds |v| < 1. Finally, we discuss the energy-critical case when p = (d + 1)/(d − 1) in dimensions d ≥ 2.
|Publication status||Published - 2018 Aug 24|
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