Time-dependent perturbation theory for abstract evolution equations of second order

A condition on a family ${\displaystyle \{B\left(t\right):t\in [0,T]\}}$ of linear operators is given under which the inhomogeneous Cauchy problem for u"(t)=(A+ B(t))u(t) + f(t) for t ∈ [0,T] has a unique solution, where A is a linear operator satisfying the conditions characterizing infinitesimal generators of cosine families except the density of their domains. The result obtained is applied to the partial differential equation $$u_{tt}=u_{xx}+b(t,x)u_x(t,x)+c(t,x)u(t,x)+f(t,x)for(t,x)\in [0,T]\times [0,1],u(t,0)=u(t,1)=0fort\in [0,T],u(0,x)=u_0(x),u_t(0,x)=v_0(x)forx\in [0,1]$$ in the space of continuous functions on [0,1].