Existence and Continuous Dependence of the Solution of Non homogeneous Wave Equation in Periodic Sobolev Spaces

• Autores: Santiago Ayala, Yolanda, Rojas Romero, Santiago
• Formato: artículo
• Fecha de Publicación: 2020
• Institución: Universidad Nacional de Trujillo
• Repositorio: Revistas - Universidad Nacional de Trujillo
• Lenguaje: español
• OAI Identifier: oai:ojs.revistas.unitru.edu.pe:article/2957
• Enlace del recurso: https://revistas.unitru.edu.pe/index.php/SSMM/article/view/2957
• Nivel de acceso: acceso abierto
• Materia: Strongly continuous operators; cosine operator; Non homogeneous wave equation; Periodic Sobolev spaces; Fourier theory; Differential Calculus in Banach Spaces; Familia de Operadores fuertemente continuos; Operador coseno; Ecuación de onda no homogénea; Espacios de Sobolev periódico; teoría de Fourier; cálculo diferencial en espacios de Banach

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.