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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

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  • $\begingroup$ You only have a mild solution for z0∈H. A solution for the differential equation exists for $z_0\in D(C)=D(A)$ only in the classical sense. $\endgroup$
    – András Bátkai
    Commented Aug 24, 2012 at 8:46

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$A$ 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.

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