Abstract
For both localized and periodic initial data, we prove local existence in classical energy space \(H^{s}, s>\frac{3}{2}\), for a class of dispersive equations \(u_t + (n(u))_x + L u_x = 0\) with nonlinearities of mild regularity. Our results are valid for symmetric Fourier multiplier operators L whose symbol is of temperate growth, and \(n(\cdot )\) in the local Sobolev space \(H^{s+2}_{\mathrm {loc}}(\mathbb {R})\). In particular, the results include non-smooth and exponentially growing nonlinearities. Our proof is based on a combination of semigroup methods and a new composition result for Besov spaces. In particular, we extend a previous result for Nemytskii operators on Besov spaces on \(\mathbb {R}\) to the periodic setting by using the difference–derivative characterization of Besov spaces.
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Both authors acknowledge the support by Grant No. 231668 by the Research Council of Norway. M.E. additionally acknowledges the support by Grant No. 250070 by the same source.
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Ehrnström, M., Pei, L. Classical well-posedness in dispersive equations with nonlinearities of mild regularity, and a composition theorem in Besov spaces. J. Evol. Equ. 18, 1147–1171 (2018). https://doi.org/10.1007/s00028-018-0435-5
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DOI: https://doi.org/10.1007/s00028-018-0435-5