Blow-up and asymptotic behavior of solutions for a class of fourth-order hyperbolic equations with mixed damping

This paper analyzes the following initial and boundary value problem of fourth-order hyperbolic equations by means of the potential well method u tt +Δ 2 u-▽u-ωΔu t +α(t)|u t | m-2 u t =|u| p-2 u, x∈Ω,t≧0, u(x,0)=u 0 (x), u t (x,0)=u 1 (x), x∈Ω u(x,t)=Δu(x,t)=0, x∈∂Ω, t≧0. Firstly, we build up ordinary differential inequalities to show the blow-up of the solution and the upper bounds for lifespan are also given in the case of initial data at different energy level. Subsequently, we discuss some results about lower bounds for lifespan with different critical case. For supercritical case, we invent the techniques which apply the interpolation inequality and energy inequalities to estabilish an inverse Hölder inequality with the correction constant. At last, we study the asymptotic behavior of global solutions in a potential well and obtain some convergence rates by using multiplier method.