Initial Boundary Value Problem For In Nonlinear Beam Equation With Strong Damping

In this paper , we present some results concerning the following nonlinear system of partial differential equationu_tt+u_xxxx-(α+βintegral from n=0 to 1(u_x²)dx)u_xx+η|u_t|^pu_t=γ|u|^qu00(1)verifying the initial condition and the boundary conditionu(0,t) = u(1,t) = u_xx(0,t) =u_xx(1,t) =0 , t≥0 (2)u(x,0) = u₀(x),u₁(x,0)= u₁(x). 0≤x≤1 (3)whereγ∈R,α,β,η,p,q >0 are constants.The particular content is following.1. We make simple comment on the developing of partial differential equation and equations relevant.2. We give some important definitions and lemmas,3. By constructing the modified potential well W associated with (1)—(3) and using a new Gronwall type integral inequality, we obtain the global weak solution for the problem(1)-(3) withγ> 0, by applying Galerkin method.4. We prove the existence and uniqueness of the global weak solution for the problem (1)-(3) withγ<0 by Galerkin method.