4In this paper,we study a nonlinear Petrovsky type equation with nonlinear weak damping,a superlinear source and time-dependent coefficients utt+△^2u-ki(t)|ut|m^2ut=k2(t)|u|^p-2u,x∈Ω,t>0,whereΩis a bounded doma...4In this paper,we study a nonlinear Petrovsky type equation with nonlinear weak damping,a superlinear source and time-dependent coefficients utt+△^2u-ki(t)|ut|m^2ut=k2(t)|u|^p-2u,x∈Ω,t>0,whereΩis a bounded domain in R^n.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 met hod 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.展开更多
基金This paper is supported by the Natural Science Foundation of Shandong Province(No.ZR2018BA016).
文摘4In this paper,we study a nonlinear Petrovsky type equation with nonlinear weak damping,a superlinear source and time-dependent coefficients utt+△^2u-ki(t)|ut|m^2ut=k2(t)|u|^p-2u,x∈Ω,t>0,whereΩis a bounded domain in R^n.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 met hod 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.
