Perturbations of an abstract Euler–Poisson–Darboux equation

The stability of the uniform correctness of the Cauchy problem
$$ \begin{gathered} u''(t)+\frac ktu'(t)=\mathbb Au(t),\qquad t>0,\qquad u(0)=u_0,\qquad u'(0)=0 \end{gathered} $$
for $k>0$ with respect to perturbations of the operator $\mathbb A$ is studied.