Global existence, uniform decay and blow-up of solutions for a system of Petrovsky equations

Abstract

In this paper we investigate global existence, uniform decay and blow-up of solutions for the following Petrovsky equations: { u t t + Δ 2 u + | u t | p − 1 u t = F u ( u , v ) , ( x , t ) ∈ Ω × [ 0 , T ) , v t t + Δ 2 v + | v t | q − 1 v t = F v ( u , v ) , ( x , t ) ∈ Ω × [ 0 , T ) , where Ω is a bounded domain of R n ( n = 1 , 2 , 3 ) having a smooth boundary and F is a C 1 function given by F ( u , v ) = α | u + v | r + 1 + 2 β | u v | r + 1 2 , r ≥ 3 , α > 1 , β > 0 . For the case of p = q = 1 , we obtain the blow-up of solutions and the lifespan estimates for four different ranges of initial energy; for the case of 1 < p , q < r , we show the blow-up of solutions when the initial energy is negative, or nonnegative at less than the mountain pass level value. Global existence of solutions is proved by the potential well theory, and decay estimates of the energy function are established by using Nakao’s inequality.