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. Nonlinear 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.

Keywords: Blow-up, viscoelastic equation, strong damping, dynamic boundary conditions