Analyzing The Radius Of Spatial Analyticity For Solutions Of Nonlinear Dispersive Partial Differential Equations

Abstract

Water waves are caused by wind that blows on the surface of water. In ocean and sea, wa- ter wave is one of natural happenings that causes considerable damage on ships, boats and other onshore activities. Thus, for human being it is natural to forecast the future disorder to reduce or overcome damages and exploit an opportunities. Different mathematical mod- els were used for such forcasting of natural phenomenon. Therefore, in this dissertation we consider the beam and fifth order Kortwege-de-Vries- Benjamin-Bona-Mahony (KdV-BBM) equations in one dimension that models weak interaction of dispersive waves and the uni- directional water waves respectively. The concept of dispersive partial differential equation which depends on the dispersion relation was explained. A new hyperbolic weight function was introduced. Using this newly introduced weight function a new modified space, which is norm equivalent to the existing Gervey space was defined. In this newly modified space the persistence of spatial analyticity for the solution of the beam equation was analyzed. In particular, for a class of analytic initial data with uniform radius of analyticity σ0 > 0, we √ obtained an asymptotic lower bound σ (t) ≤ c/ t on the uniform radius of analyticity σ (t) of solution as time t goes to infinity. Also for the fifth-order KdV-BBM model with analytic initial data and fixed radius σ0 , we proved that the radius of spatial analyticity can not decay √ faster than 1/ T for any large time T .