Abstract
The aim of this paper is to study bounds for blow-up time to the following viscoelastic hyperbolic equation of Kirchhoff type with initial-boundary value condition: $$ |u_{t}|^{\rho }u_{tt}-M(\|\nabla u\|_{2}^{2})\Delta u+\int _{0}^{t}g(t- \tau )\Delta u(\tau )d\tau +|u_{t}|^{m(x)-2}u_{t}=|u|^{p(x)-2}u. $$ Compared with constant exponents, it is difficult to discuss the above problem due to the existence of a gap between the modular and the norm. The authors construct suitable function spaces to discuss the upper bound for blow-up time with positive initial energy by means of a differential inequality technique. In addition, lower bounds for blow-up time in different range of exponent are obtained. These improve and generalize some recent results.
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