Global existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source interaction

Said-Houari, Belkacem; Nascimento, Flávio A Falcão

Global existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source interaction

Type

Article

Authors

Said-Houari, Belkacem
Nascimento, Flávio A Falcão

KAUST Department

Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Date

2012-09

Abstract

The goal of this work is to study a model of the viscoelastic wave equation with nonlinear boundary/interior sources and a nonlinear interior damping. First, applying the Faedo-Galerkin approximations combined with the compactness method to obtain existence of regular global solutions to an auxiliary problem with globally Lipschitz source terms and with initial data in the potential well. It is important to emphasize that it is not possible to consider density arguments to pass from regular to weak solutions if one considers regular solutions of our problem where the source terms are locally Lipschitz functions. To overcome this difficulty, we use an approximation method involving truncated sources and adapting the ideas in [13] to show that the existence of weak solutions can still be obtained for our problem. Second, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term, then the solution ceases to exist and blows up in finite time provided that the initial data are large enough.

Citation

Said-Houari, B., & Nascimento, F. A. F. (2012). Global existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source interaction. Communications on Pure and Applied Analysis, 12(1), 375–403. doi:10.3934/cpaa.2013.12.375

Acknowledgements

Doctorate student by State University of Maringa, partially supported by a grant of CNPq, BrazilThe authors thanks Prof. Marcelo Moreira Cavalcanti for many helpful comments, which improve the first version of this paper. Moreover, the first author thanks KAUST for its support.