The existence and decay of solutions of a damped Kirchhoff–Carrier equation in Banach spaces

This paper is concerned with the study of the existence and decay of solutions of the following initial value problem: equation(∗)|Bu′′(t)+M(‖u(t)‖Wβ)Au(t)+(1+k(t)‖u(t)‖D(Sα+2)β)Au′(t)=0,t>0u(0)=u0,u′(0)=u1, where VV is a Hilbert space with dual V′V′; AA and BB symmetric linear operators from VV into V′V′ with 〈Bv,v〉>0〈Bv,v〉>0, v≠0v≠0, and 〈Av,v〉≥γ‖v‖V2, γ>0γ>0; SS a restriction of the operator A;WA;W a Banach space; M(ξ)M(ξ) the real function M(ξ)=m0+m1ξM(ξ)=m0+m1ξ with m0>0m0>0 and m1≥0m1≥0 real numbers; kk a positive function and α,βα,β real numbers with α≥0α≥0 and β>1β>1.The successive approximation method, the characterization of the derivative of M(‖u(t)‖Wβ) and the Arzela–Áscoli Theorem allow us to obtain a local solution of (∗). The global solution follows by the prolongation method of solutions. The exponential decay of the solution is derived by the perturbed energy method.