Classical well-posedness in dispersive equations with nonlinearities of mild regularity, and a composition theorem in Besov spaces

Abstract

For both localized and periodic initial data, we prove local existence in classical energy space Hs,s>32, for a class of dispersive equations ut+ (n(u)) x+ Lux= 0 with nonlinearities of mild regularity. Our results are valid for symmetric Fourier multiplier operators L whose symbol is of temperate growth, and n(·) in the local Sobolev space Hlocs+2(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 R to the periodic setting by using the difference–derivative characterization of Besov spaces.