【作者】 游淑军；

【导师】 尚亚东；

【作者基本信息】 广州大学 ， 应用数学， 2006， 硕士

【摘要】 本文讨论一类广义Zakharov方程组.分别在一维情形和二维情形下,研究该方程组的周期初值问题整体古典解的存在性与唯一性.并且在一维情形下,研究该方程组局部解的blow-up问题.存在唯一性的证明方法是先作Gale¨rkin近似解,得到局部古典解,再由积分先验估计得到整体解.证明了:在一维情形下,当0 < p < 4时,方程的周期初值问题的整体古典解是存在且唯一的;在二维情形下当0 < p < 2及ε0(x)充分小时,方程的初值问题的整体古典解是存在且唯一的.并且在一维情形下得到了:当p≥2(1+πa2E1?0)π+a22√E10+ 3πa2E0及初值满足一定条件时,方程的解在有限时间内出现blow-up.更多还原

【Abstract】 In this paper, we study a class of generalized Zakharov equations. We considerexistence and uniqueness of the solutions of the periodic initial value problem of thisequations in one dimension space and in two dimension space. The blow-up of the lo-cal solutions of the equations is also discussed. By means of the Gale¨rkin method andthe integral estimates, we obtain the following conclusions: In one dimension space,if 0 < p < 4, the existence and uniqueness of the global classical solution is obtained;In two dimension space, if 0 < p < 2 andε0(x) is enough small, the existenceand uniqueness of the global classical solution is obtained. Then, in one dimensionspace, if p≥2(1+πa2E1?0)π+a22√E10+ 3πa2E0and initial data satisfy some conditions, the localsolution will blow up in finite time.更多还原