Journal of the
Korean Mathematical Society JKMS

In this paper we consider the following strongly damped wave equation with variable-exponent nonlinearity $$u_{tt}(x,t) - \Delta u (x,t) - \Delta u_t (x,t) = |u(x,t)|^{p(x)-2} u(x,t) , $$ where the exponent $p(\cdot)$ of nonlinearity is a given measurable function. We establish finite time blow-up results for the solutions with non-positive initial energy and for certain solutions with positive initial energy. We extend the previous results for strongly damped wave equations with constant exponent nonlinearity to the equations with variable-exponent nonlinearity.

Keywords: Wave equation, variable-exponent nonlinearity, finite time blow-up, strong damping

MSC numbers: 35L05, 35L70, 35B44

Supported by: This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2020R1I1A3066250)