On decay properties for solutions of the Zakharov-Kuznetsov equation

Abstract

This work mainly focuses on the spatial decay properties of solutions to the Zakharov-Kuznetsov equation. In earlier studies for the two- and three-dimensional cases, it was established that if the initial condition $u_0$ verifies $\langle \sigma\cdot x\rangle^{r}u_{0}\in L^{2}(\left\{\sigma\cdot x\geq \kappa\right\}),$ for some $r\in\mathbb{N}$, $\kappa \in\mathbb{R}$, being $\sigma$ be a suitable non-null vector in the Euclidean space, then the corresponding solution $u(t)$ generated from this initial condition verifies $\langle \sigma\cdot x\rangle ^{r}u(t)\in L^2\left(\left\{\sigma\cdot x>\kappa-\nu t\right\}\right)$, for any $\nu >0$. 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 in terms of the magnitude of the weight $r$. The deduction of our results depends on a new class of pseudo-differential operators, which is useful to quantify 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, $e^{b\, \sigma\cdot x}u_{0}\in L^{2}(\left\{\sigma\cdot x\geq \kappa\right\}),$ then the corresponding solution satisfies $e^{b\, \sigma\cdot x} u(t)\in H^{p}\left(\left\{\sigma\cdot x>\kappa-\nu t\right\}\right),$ for all $p\in\mathbb{N}$, and time $t\geq \delta,$ 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.