This paper deals with the following Petrovsky equation with damping and nonlinear sources:utt+△^(2)u-M(||■u||2^(2))△ut+|ut|^(m(x)-2)ut=|u|^(p(x)-2)u under initial-boundary value conditions,where M(s)=a+b sγis a po...This paper deals with the following Petrovsky equation with damping and nonlinear sources:utt+△^(2)u-M(||■u||2^(2))△ut+|ut|^(m(x)-2)ut=|u|^(p(x)-2)u under initial-boundary value conditions,where M(s)=a+b sγis a positive C 1 function with the parameters a>0,b>0,γ≥1,and m(x)and p(x)are given measurable functions.The upper bound of the blow-up time is derived for low initial energy by the differential inequality technique.For m(x)≡2,in particular,the upper bound of the blow-up time is obtained by the combination of Levine's concavity method and some differential inequalities under high initial energy.In addition,we discuss the lower bound of the blow-up time by making full use of the strong damping.Moreover,we present the global existence of solutions and an energy decay estimate by establishing some energy estimates.展开更多
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supported by National Natural Science Foundation of China(Grant No.12071391)。
文摘This paper deals with the following Petrovsky equation with damping and nonlinear sources:utt+△^(2)u-M(||■u||2^(2))△ut+|ut|^(m(x)-2)ut=|u|^(p(x)-2)u under initial-boundary value conditions,where M(s)=a+b sγis a positive C 1 function with the parameters a>0,b>0,γ≥1,and m(x)and p(x)are given measurable functions.The upper bound of the blow-up time is derived for low initial energy by the differential inequality technique.For m(x)≡2,in particular,the upper bound of the blow-up time is obtained by the combination of Levine's concavity method and some differential inequalities under high initial energy.In addition,we discuss the lower bound of the blow-up time by making full use of the strong damping.Moreover,we present the global existence of solutions and an energy decay estimate by establishing some energy estimates.
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.
