Existence and Continuous Dependence of the Local Solution of Non Homogeneous Third Order Equation and Generalizations

In this article, we prove that initial value problem associated to the non homogeneous third order equation in periodic Sobolev spaces has a local so- lution in [0, T ] with T > 0, and the solution has continuous dependence with respect to the initial data and the non homogeneous part of the problem. We do this in a intuitive way using Fourier theory and introducing a Co - Semi- group inspired by the work of Iorio [1] and Santiago [6]. Also, we prove the uniqueness solution of the homogeneous third order equa- tion, using its conservative property, inspired by the work of Iorio [1] and Santiago [7]. Finally, we study its generalization to n-th order equation.