Upper and lower bounds for the blow-up time for a viscoelastic wave equation with dynamic boundary conditions

Abstract

In this paper, we consider a viscoelastic wave equation with dynamical boundary conditions. Under certain assumptions, we give the upper and lower bounds for the blow-up time according to the exponent numbers m and p of the nonlinear boundary damping term and the source term. For the case 2 ≤ m < p, we extend the earlier exponentially growth result in Gerbi and Said-Houari (Adv. Non-linear Analysis, 2013) to a blow-up in finite time result with positive initial energy and get the upper bound for the blow-up time. For the case m = 2, by using the concavity method, we prove a finite time blow-up result and get the upper bound for the blow-up time, which is a supplement to Gerbi and Said-Houari (Adv. Nonlinear Analysis, 2013). Moreover, for the case m ≥ 2, under certain conditions on the data, we give a lower bound for the blow-up time when blow-up occurs.

Mathematics Subject Classification (2010):

• 35B44
• 35L05
• 35L15
• 35L70

Key words:

• Blow-up
• viscoelastic equation
• strong damping
• dynamic boundary conditions