On decay properties for solutions of the Zakharov-Kuznetsov equation

This work mainly focuses on spatial decay properties of solutions to the Zakharov-Kuznetsov equation. For the two- and three-dimensional cases, it was established that if the initial condition u0 verifies <sigma & sdot;x >(R) u(0)is an element of L-2(sigma & sdot;x >=kappa), for some r is an element of N, kappa is an element of R, being sigma be a suitable non-null vector in the Euclidean space, then the corresponding solution u(t) generated from this initial condition verifies <sigma & sdot;x >(R) u(t)is an element of L-2 sigma & sdot;x>kappa-nu t, for any nu>0. Additionally, depending on the magnitude of the weight r, it was also deduced some localized gain of regularity. In this regard, we first extend such results to arbitrary dimensions, decay power r>0 not necessarily an integer, and we give a detailed description of the gain of regularity propagated by solutions. The deduction of our results depends on a new class of pseudo-differential operators, which is useful for quantifying decay and smoothness properties on a fractional scale. Secondly, we show that if the initial data u(0) has a decay of exponential type on a particular half space, that is, e(b sigma & sdot;x)u(0)is an element of L-2(sigma & sdot;x >=kappa), then the corresponding solution satisfies e(b sigma & sdot;x)u(t)is an element of H-p sigma & sdot;x>kappa-t, for all p is an element of N, and time t >=delta, where delta>0. To our knowledge, this is the first study of such property. As a further consequence, we also obtain well-posedness results in anisotropic weighted Sobolev spaces in arbitrary dimensions. Finally, as a by-product of the techniques considered here, we show that our results are also valid for solutions of the Korteweg-de Vries equation.