Abstract
We consider the focusing L 2-critical half-wave equation in one space dimension,
, where D denotes the first-order fractional derivative. Standard arguments show that there is a critical threshold \({M_{*} > 0}\) such that all H 1/2 solutions with \({\|u\|_{L^2} < M_*}\) extend globally in time, while solutions with \({\|u\|_{L^2} \geq M_*}\) may develop singularities in finite time. In this paper, we first prove the existence of a family of traveling waves with subcritical arbitrarily small mass. We then give a second example of nondispersive dynamics and show the existence of finite-time blowup solutions with minimal mass \({\|u_0\|_{L^2} = M_*}\). More precisely, we construct a family of minimal mass blowup solutions that are parametrized by the energy E 0 > 0 and the linear momentum \({P_0 \in \mathbb{R}}\). In particular, our main result (and its proof) can be seen as a model scenario of minimal mass blowup for L 2-critical nonlinear PDEs with nonlocal dispersion.
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Krieger, J., Lenzmann, E. & Raphaël, P. Nondispersive solutions to the L 2-critical Half-Wave Equation. Arch Rational Mech Anal 209, 61–129 (2013). https://doi.org/10.1007/s00205-013-0620-1
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DOI: https://doi.org/10.1007/s00205-013-0620-1