Oscillation For Nolinear Neutral Delay Partial Differential Equations With Continuous Distributed Delays

The study of oscillatory behavior of solutions of partial differential equations with continuous distributed delays, besides its theoretical interest, is important from the viewpoint of applications. The oscillation of partial differential equation has recently been investigated by many authors. The most studies are about the partial differential equation with discrete delays, but the attention to the partial differential equation with continuous distributed delays is much less. In this paper, the oscillation of the partial differential equation with continuous distributed delays is studied. Some new oscillation criteria are given for a class of nonlinear two order neutral partial differential equation with continuous distributed delays and a class of nonlinear higher order neutral partial differential equation with continuous distributed delays.The two classes of equations are both with the same three kinds of boundary conditions. The whole paper is divided into three setions.In section one , correlative concepts about the theory of the oscillation of the partial differential equation , the history of the theory and the rescent advance in the study are introduced, as well as the work we do on the problems.In section two, we discuss the oscillation of nonlinear two order neutral partial differential equation with continuous distributed delays under the three kinds of boundary coditions. Basing on this ,we divide the section into three parts, and prove respectively several oscillations of the equation with different initial values. The main methods here used are direct integrating , Green's formula , Jensen's inequality. We transform the partial differential equation into differential eqution firstly, then use the Riccati transform and some lemmas to get the oscillations of the equations.In section three ,we discuss the oscillation of nonlinear higher order neutral partial differential equation with continuous distributed delays. Here we generalize the two order equation in section two to higher order equation. As the same three kinds of boundary coditions, we divide the section into three parts, and prove respectively several oscillations of the equation with different initial values. The main methods here used are direct integrating , Green's formula , Jensen's inequality. We transform the partial differential equation into differential eqution firstly, then use the Riccati transform and some lemmas to get the oscillations of the equations.