Behavior of solutions to a Petrovsky equation with damping and variable-exponent sources

This paper deals with the following Petrovsky equation damping and nonlinear sources: $${u_{tt}} + {\Delta ^2}u - M\left({\left\| {\nabla u} \right\|_2^2} \right)\Delta u \Delta {u_t} {\left| {{u_t}} \right|^{m(x) 2}}{u_t} = \right|^{p(x) 2}}u$$ under initial-boundary value conditions, where M(s) a bsγ is positive C1 function parameters > 0, b γ ⩾ 1, m(x) p(x) are given measurable functions. The upper bound of blow-up time derived for low initial energy by differential inequality technique. For ≡ 2, in particular, obtained combination Levine’s concavity method some inequalities high energy. In addition, we discuss lower making full use strong damping. Moreover, present global existence solutions an decay estimate establishing estimates.