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

• Main Authors: Yolanda Santiago Ayala, Santiago Rojas Romero
• Format: Article
• Language: Spanish
• Published: Universidad Nacional de Trujillo 2020-07-01
• Series: Selecciones Matemáticas
• Subjects: strongly continuous operators; cosine operator; non homogeneous wave equation; periodic sobolev spaces; fourier theory; differential calculus in banach spaces
• Online Access: https://revistas.unitru.edu.pe/index.php/SSMM/article/view/2957/3286

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.