Time-dependent perturbation theory for abstract evolution equations of second order
Volume 130 / 1998
Studia Mathematica 130 (1998), 263-274
DOI: 10.4064/sm-130-3-263-274
Abstract
A condition on a family ${B(t):t ∈ [0,T]}$ 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 $$ u_{tt} = u_{xx} + b(t,x)u_x(t,x) + c(t,x)u(t,x) + f(t,x) for (t,x) ∈ [0,T]×[0,1], u(t,0) = u(t,1) = 0 for t ∈ [0,T], u(0,x) = u_0(x), u_t(0,x) = v_0(x) for x ∈ [0,1] $$ in the space of continuous functions on [0,1].
