Existencia y Dependencia Continua de la Solución de la Ecuación de Onda no Homogénea en Espacios de Sobolev Periódico

Abstract

In this article, we first prove that the initial value problem associated to the homogeneous wave equation in periodic Sobolev spaces has a global solution and the solution has continuous dependence with respect to the initial data, in [0; T], T > 0. We do this in an intuitive way using Fourier theory and in a fine version introducing families of strongly continuous operators, inspired by the works of Iorio [4] and Santiago and Rojas [7]. Also, we prove that the energy associated to the wave equation is conservative in intervals [0; T], T > 0. As a final result, we prove that the initial value problem associated to the non homogeneous wave equation has a local solution and the solution has continuous dependence with respect to the initial data and the non homogeneous part of the problem.

Keywords

• strongly continuous operators
• cosine operator
• non homogeneous wave equation
• periodic sobolev spaces
• fourier theory
• differential calculus in banach spaces