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- W4321853961 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 $langle sigmacdot xrangle^{r}u_{0}in L^{2}(left{sigmacdot xgeq kapparight}),$ for some $rinmathbb{N}$, $kappa inmathbb{R}$, being $sigma$ be a suitable non-null vector in the Euclidean space, then the corresponding solution $u(t)$ generated from this initial condition verifies $langle sigmacdot xrangle ^{r}u(t)in L^2left(left{sigmacdot x>kappa-nu tright}right)$, 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, $e^{b, sigmacdot x}u_{0}in L^{2}(left{sigmacdot xgeq kapparight}),$ then the corresponding solution satisfies $e^{b, sigmacdot x} u(t)in H^{p}left(left{sigmacdot x>kappa-nu tright}right),$ for all $pinmathbb{N}$, and time $tgeq delta,$ 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." @default.
- W4321853961 created "2023-02-25" @default.
- W4321853961 creator A5002556583 @default.
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- W4321853961 date "2023-02-22" @default.
- W4321853961 modified "2023-10-14" @default.
- W4321853961 title "On decay properties for solutions of the Zakharov-Kuznetsov equation" @default.
- W4321853961 doi "https://doi.org/10.48550/arxiv.2302.11731" @default.
- W4321853961 hasPublicationYear "2023" @default.
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