We study the initial-boundary value problem for a nonlinear wave equation given by u_{ tt}-u_{ xx}+\int_{ 0}^{ t}k(t-s)u_{ xx}(s)ds+ u_{ t}^{ q-2}u_{ t}=f(x,t,u) , 0 < x < 1, 0 < t < T, u_{ x}(0,t)=u(0,t), u_{ x}(1,t)+\eta u(1,t)=g(t), u(x,0)=\^u_{ 0}(x), u_{ t}(x,0)={ \^u}_{ 1}(x), where \eta \geq 0, q\geq 2 are given constants { \^u}_{ 0}, { \^u}_{ 1}, g, k, f are given functions. In part I under a certain local Lipschitzian condition on f, a global existence and uniqueness theorem is proved. The proof is based on the paper [10] associated to a contraction mapping theorem and standard arguments of density. In Part} 2, under more restrictive conditions it is proved that the solution u(t) and its derivative u_{ x}(t) decay exponentially to 0 as t tends to infinity.
Categories: math.AP