Remarks on a Nonlinear Wave Equation in a Noncylindrical Domain

Ivo Fernandez Lopez, Gladson Octaviano Antunes, Maria Darci Godinho da Silva, Luiz Adauto da Justa Medeiros

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. 


Keywords


Nonlinear problem; Non-cylindrical domain; Hyperbolic equation

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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.




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

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