Remarks on a Nonlinear Wave Equation in a Noncylindrical Domain
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.
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Lions, J. L., “Quelques Méthodes de Résolution des Problemes aux Limites Non Linéaires”, Dunod, Paris, 1969.
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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.
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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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Trends in Computational and Applied Mathematics
A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)
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