Remarks on a Nonlinear Wave Equation in a Noncylindrical Domain
DOI:
https://doi.org/10.5540/tema.2015.016.03.0195Keywords:
Nonlinear problem, Non-cylindrical domain, Hyperbolic equationAbstract
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.
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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