Global solution and exponential decay for a nonlinear hyperbolic equation in Banach spaces

In this work we study the existence, uniqueness and asymptotic behavior, as t→∞t→∞, of solutions of the initial value problem {u″(t)+M(t,‖u(t)‖Wβ)Au(t)+F(u(t))+(1+α‖u(t)‖Vβ)Au′(t)=0in H,u(0)=u0,u′(0)=u1, where MM is a function satisfying suitable conditions, AA and FF are operators, VV and HH are two Hilbert spaces, WW is a Banach space, α>0α>0, β≥2β≥2 are two real numbers and u0u0, u1u1 are initial data. The global solutions is obtained by use of Faedo–Galerkin’s method together with a characterization of the derivative of the nonlinear term M(t,‖u(t)‖Wβ) and the Arzelà–Ascoli theorem. The exponential decay of solutions is analyzed by means of the perturbed energy method.