On The Radius of Analyticity of Solutions to Some Dispersive Equations

Getachew, Tegegne

dc.contributor.author	Getachew, Tegegne
dc.date.accessioned	2024-07-24T11:05:16Z
dc.date.available	2024-07-24T11:05:16Z
dc.date.issued	2024-07
dc.identifier.uri	http://ir.bdu.edu.et/handle/123456789/15900
dc.description.abstract	In this dissertation, we studied the well-posedness and decay rate on the radius of spatial analyticity for solutions to some class of nonlinear dispersive PDEs applicable in many problems involving water waves. In particular, our study focused on the nonlinear Schrodinger equation, fourth-order Schr ¨ odinger equation with cubic nonlinearity, gen- ¨ eralized Korteweg de-Vries equation, generalized Korteweg de-Vries equations with higher-order dispersion and nonlinear wave equation with initial data in the modified Gevrey space. We first introduce the initial value problems for each models. Second, we study the local-in-time well-posedness of them using different estimates and the stan dard Banach’s fixed point principle. Third, we construct modified mass and energy (or energy) functionals, which are shown to be almost conserved. Fourth, we differentiate those mass and energy (or energy) functionals, and then applying different space time estimates, Plancherel’s Theorem, real interpolation, Minkowski’s inequalities, Young’s inequality and one dimensional Sobolev embedding to prove corresponding almost (ap proximate) conservation law. Finally, the iteration of the almost conservation law for the modified mass and energy (or energy) functionals over time intervals of uniform length combined with the local well-posedness can extend the solution to global in time. We further estimate the decay-in-time-rate on the radius of spatial analyticity. In the fourth chapter, we presented our first result and show that the uniform radius of spatial analyticity σ(t) of the solution to the one-dimensional second-order NLS equation is bounded from below by c\|t\| − 2 3 when the nonlinearity is of any odd natural number greater than three and by c\|t\| − 4 5 when the nonlinearity is cubic for large \|t\| > 0, given initial data that is analytic with fixed radius σ0. This improves the results obtained by Ahn et al. and Tesfahun, respectively, where they obtained a decay rate of order \|t\| −1 . In the fifth chapter, we showed that the fourth order Schrodinger equation with cubic ¨ nonlinearity is locally and globally well-posed in the modified Gevrey space. We have proved that the uniform radius of spatial analyticity σ(t) of solution at time t cannot decay faster than t − 1 2 for large t, given initial data that is analytic with fixed radius σ0. In the sixth chapter, we obtained the decay rate for the radius of spatial analyticity of solutions to the generalized Korteweg de-Vries equation. We have shown that the analyticity radius does not decay faster than \|t\| − 5 6 if the nonlinearity is cubic and \|t\| − 3 5 if the nonlinearity is the power of any odd natural number greater than three for large time t, which improves earlier results due to Bona, Grujic and Kalisch. ´ vi In the seventh chapter, we studied the Cauchy problem for the defocusing gener alized KdV equation with any odd order of dispersion greater than or equal to five. It is shown that, given analytic data with a fixed radius σ0, the uniform radius of spatial analyticity σ(t) of solutions at time t cannot decay faster than \|t\| − 1 2 for large time t. In particular, this improves a recent result due to Petronilho and Silva for the modified Kawahara equation (the dispersion order is five and of cubic nonlinearity), where they obtained a decay rate of order t −4+. In the eighth chapter, we analyzed the decay rate for the radius of spatial analyticity for solutions of the nonlinear wave equation, subject to initial data, which is analytic in modified Gevrey space with a fixed radius σ0. It is shown that the asymptotic lower bound σ(t) > c\|t\| − 2 3 as \|t\| → +∞. This is an improvement of a recent result by da Silva and Castro, where they obtained a decay rate of order (1 + \|t\|) −( p+1 2 ) for any odd natural number p > 3.	en_US
dc.language.iso	en_US	en_US
dc.subject	Mathematics	en_US
dc.title	On The Radius of Analyticity of Solutions to Some Dispersive Equations	en_US
dc.type	Dissartation	en_US