Wellposedness Of A Class Of Generalized Zakharov Equations

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.