Let $A$ a linear operator from his domaine $D(A)\subset H$ to $H$, with $H$ is a Hilbert space, such that $A$ is dissipative, and let $B$ is a monotone linear operator such that $D(A)\subset D(B)$.
we suppose that $C=A-B$ gererate a contraction semi group $T(t)$ on $H$; then for all $z_0\in H$ $\frac{dz}{dt}=Cz$ admet a unique solution in H.
Question :
under what conditions we have $t\longrightarrow\|Az(t)\|$ is an increasing function ?
Thank you
$A$ is with domain, so the norm of $A$ may be infinite. Also, one can derivite along the trajectory provivded that the initial data is taken in $D(A)$ and perhaps under additional assumptions on the perturbation $B$. In this case if $A$ and $B$ commutes and $B\ge 0,$ then based on the reply of Daniel and the remark of Willie we get the conclusion. Now to extend the result to $z_0\in H,$ one can use the density of $D(A)$ in $H$ provided the solution $z(t)$ depends continuously on the initial state.
