Blow up of Solutions for a Nonlinear Petrovsky Type Equation with Time-dependent Coefficients

In this paper, we study a nonlinear Petrovsky type equation with nonlinear weak damping, a superlinear source and time-dependent coefficients $${u_{tt}} + {{\rm{\Delta }}^2}u + {k_1}\left(t \right){\left| {{u_t}} \right|^{m - 2}}{u_t} = {k_2}\left(t \right){\left| u \right|^{p - 2}}u,\,\,\,\,\,\,\,\,\,\,x \in {\rm{\Omega,}}\,\,\,{\rm{t > 0,}}$$ where Ω is a bounded domain in Rn. Under certain conditions on k1(t), k2(t) and the initial-boundary data, the upper bound for blow-up time of the solution with negative initial energy function is given by means of an auxiliary functional and an energy estimate method if p > m. Also, a lower bound of blow-up time are obtained by using a Sobolev-type inequality and a first order differential inequality technique for n = 2, 3 and n > 4.