Remarks on a Nonlinear Wave Equation in a Noncylindrical Domain

Authors

  • Ivo Fernandez Lopez Universidade Federal do Rio de Janeiro
  • Gladson Octaviano Antunes Universidade Federal do Estado do Rio de Janeiro
  • Maria Darci Godinho da Silva Universidade Federal do Rio de Janeiro
  • Luiz Adauto da Justa Medeiros Universidade Federal do Rio de Janeiro

DOI:

https://doi.org/10.5540/tema.2015.016.03.0195

Keywords:

Nonlinear problem, Non-cylindrical domain, Hyperbolic equation

Abstract

In this paper we investigate the existence and uniqueness of solution for a initial boundary value problem for the following nonlinear wave equation:

u′′ − ∆ u + | u | ˆρ = f in Q

where Q represents a non-cylindrical domain of R^{n+1}. The methodology, cf. Lions [3], consists of transforming this problem, by means of a perturbation depending on a parameter ε > 0, into another one defined in a cylindrical domain Q containing Q. By solving the cylindrical problem, we obtain estimates that depend on ε. These ones will enable a passage to the limit, when ε goes to zero, that will guarantee, later, a solution for the non-cylindrical problem. The nonlinearity |u_ε|^ρ introduces some obstacles in the process of obtaining a priori estimates and we overcome this difficulty by employing an argument due to Tartar [8] plus a contradiction process. 

Author Biographies

Ivo Fernandez Lopez, Universidade Federal do Rio de Janeiro

Departamento de Métodos Matemáticos

Gladson Octaviano Antunes, Universidade Federal do Estado do Rio de Janeiro

Departamento de Matemática e Estatística

Maria Darci Godinho da Silva, Universidade Federal do Rio de Janeiro

Departamento de Métodos Matemáticos

Luiz Adauto da Justa Medeiros, Universidade Federal do Rio de Janeiro

Departamento de Métodos Matemáticos

References

Lions, J. L., “Quelques Méthodes de Résolution des Problemes aux Limites Non Linéaires”, Dunod, Paris, 1969.

Lions, J. L., Magenes, E., “Non-homogeneous Boundary Value Problems and Applications”, Spring-Verlag, New York, 1972.

Lions, J. L.,Strauss, W. A., Some nonlinear evolution equations, Bol. Soc. Math. de France, 93, pp. 43-96, 1965.

Medeiros, L. A., Límaco, J. and Frota, C. L., On wave equations without global a priori estimates, Bol. Soc. Paranaense de Matemática, 30-2, pp. 12-32, 2012.

Satinger, D. H., On global solutions for nonlinear hyperbolic equations, Arch. Rational Mech. and Analysis, 30, pp. 148-172, 1968.

Tartar, L., “Topics in Nonlinear Analysis”, Un. Paris Sud. Dep. Math., Orsay, France, 1978.

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Published

2016-01-28

How to Cite

Lopez, I. F., Antunes, G. O., da Silva, M. D. G., & Medeiros, L. A. da J. (2016). Remarks on a Nonlinear Wave Equation in a Noncylindrical Domain. Trends in Computational and Applied Mathematics, 16(3), 195. https://doi.org/10.5540/tema.2015.016.03.0195

Issue

Vol. 16 No. 3 (2015)

Section

Original Article

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