Global existence and exponential growth for a viscoelastic wave equation with dynamic boundary conditions.

Stéphane Gerbi (1) , Belkacem Said-Houari (2, 3)

1 LAMA - Laboratoire de Mathématiques
2 UBMA - Université Badji Mokhtar [Annaba]
3 MCSE - Division of Mathematical and Computer Sciences and Engineering

The goal of this work is to study a model of the wave equation with dynamic boundary conditions and a viscoelastic term. First, applying the Faedo-Galerkin method combined with the fixed point theorem, we show the existence and uniqueness of a local in time solution. Second, we show that under some restrictions on the initial data, the solution continues to exist globally in time. On the other hand, if the interior source dominates the boundary damping, then the solution is unbounded and grows as an exponential function. In addition, in the absence of the strong damping, then the solution ceases to exist and blows up in finite time.

Mots clés: Damped viscoelastic wave equations; global solutions; exponential growth; blow up in finite time; dynamic boundary conditions.; dynamic boundary conditions

Domaines: Equations aux dérivées partielles [math.AP]

Stéphane Gerbi, Belkacem Said-Houari. Global existence and exponential growth for a viscoelastic wave equation with dynamic boundary conditions.. Advances in Nonlinear Analysis, 2013, 2 (2), pp.163-193. ⟨10.1515/anona-2012-0027⟩. ⟨hal-00789174⟩