Is there an analogue of the Hille Yosida theorem for abstract Cauchy problems of the type $ \frac{du(x,t)}{dt}=A(t)u(t,x)+Cu(t,x)$

Question

I have the following abstract Cauchy problem on a Banach space $X$: $$ \frac{du(x,t)}{dt}=A(t)u(t,x)+Cu(t,x)$$ where $A(t)$ is a closed, densely defined, linear, bounded operator for every $t \in [0,T]$.

Moreover, all the operators of the family $\{A(t): t \in [0,T]\}$ have the same domain $D(A) \subset X $, hence all the operators go from $D(A)$ to $X$. Also $C$ is a linear bounded operator from $D(C)$ to $X$.

I would like to know if the operator $A(t)+C$ generates a strongly continuous semigroup. I know that this holds if $A$ does not depend by the time, but I do not find the analogue result in the case of $A$ depending by time.

Someone knows where this kind of problems are treated in literature ? Thank you

Answer 1

As $A$ depends on $t$, we have to talk about "family" of infinitesimal generators. In this context, your question can be formulated as follows:

Does a family of generators retain the properties of a family of generators if subjected to perturbations? (This is a version of Kato, p. 497)

Of course, the answer depends on the properties of the family and the perturbations. We have, for example, the following result in Pazy's book (p. 132):

If $\{A(t)\}_{t\in[0,T]}$ is a stable family of infinitesimal generators of $C_0$-semigroups and $\{B(t)\}_{t\in[0,T]}$ is a family of bounded linear operators from $X$ to $X$ such that $\sup_{t\in[0,T]}\|B(t)\|<\infty$, then $\{A(t)+B(t)\}_{t\in[0,T]}$ is a stable family of infinitesimal generators.

You can find more results in the paper Stability of CD-Systems Under Perturbations in the Favard class (there are other references in the third paragraph).