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

Authors

DOI:

https://doi.org/10.17268/sel.mat.2020.01.06

Keywords:

Strongly continuous operators, cosine operator, Non homogeneous wave equation, Periodic Sobolev spaces, Fourier theory, Differential Calculus in Banach Spaces

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.

Author Biography

Santiago Rojas Romero, Facultad de Ciencias Matemáticas, Universidad Nacional Mayor de San Marcos, Av. Venezuela S/N Lima 01, Lima-Perú.

Facultad de Ciencias Matemáticas, Universidad Nacional Mayor de San Marcos, Av. Venezuela S/N Lima 01, Lima-Perú.

References

Benjamin T. Internal Waves of Permanent Form in Fluids of Great Depth. J. Fluid Meek. 1967; 29: 559-592.

Courant R, Hilbert D. Methods of Mathematical Physics. Interscience, Wile, New York. 1962; 2.

Folland G, Sitaram A. The uncertainty principle: a mathematical survey. J. Fourier Anal. Appl. 1997; 3: 207-238.

Iorio R, Iorio V. Fourier Analysis and partial differential equation. Cambridge University, 2001.

Santiago Y, Rojas S, Quispe T. Espacios de Sobolev periódico y un problema de Cauchy asociado a un modelo de ondas en un fluido viscoso. Theorema, Segunda ´ Epoca. 2016; 3(4): 7-23.

Santiago Y, Rojas S. Existencia y Regularidad de solución de la ecuaci´on del calor en espacios de Sobolev Periódico. Selecciones Matemática. 2019; 06(01): 49-65.

Santiago Y, Rojas S. Regularity and wellposedness of a problem to one parameter and its behavior at the limit. Bulletin of the Allahabad Mathematical Society. 2017; 32(2): 207-230.

Published

2020-07-25

How to Cite

Santiago Ayala, Y., & Rojas Romero, S. (2020). Existence and Continuous Dependence of the Solution of Non homogeneous Wave Equation in Periodic Sobolev Spaces. Selecciones Matemáticas, 7(01), 52-73. https://doi.org/10.17268/sel.mat.2020.01.06

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