I asked this question on [Mathematics Stackexchange](https://math.stackexchange.com/questions/3589223/u-t-aufu-where-a-is-the-infinitesimal-generator-of-c-0-semigroup), but got no answer. 


In Pavel's book: Nonlinear Evolution Operators and Semigroups - Applications to Partial Differential Equations, we have the following definitions and result:


Consider the following semilinear equation:


$$\tag{5.59}\begin{cases}u\_t(t)=Au(t)+F(u(t)), ~0 \leq t \leq T,\\u(0)=u\_0 \in H. \end{cases} $$
where $A:D(A)\subset H \rightarrow H$ be the infinitesimal generator of $C\_0$-semigroup $S(t)$ with $\|S(t)\|\leq 1$.



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> **Definition 5.1.** (1)A function $u:[0,T]\rightarrow H$ is said to be a classical solution of $(5.59)$ if $u(0)=u\_0$ and $$\tag{5.68}u \in C([0,T];D(A))\cap C^1([0,T];H)$$
>  and $(5.59)$ is satisfied. (2) The function $u$ is a classical solution on $[0,\infty[$ of $(5.59)$ if $(5.68)$ holds for every $T>0$ and $(5.59)$ is satisfied for every $t>0$.
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By a mild solution of $(5.59)$ we mean a continuous function $u:[0,T] \rightarrow H$ satisfying $$\tag{5.69} u(t)=S(t)u\_0+\int\_{0}^{t} S(t-s)F(u(s))\, ds, ~0 \leq t \leq T.$$



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> **Theorem 5.8** If $F:H\rightarrow H$ is locally Lipschitz then for every $u \in H$, there is a unique mild solution $u:[0,T\_{\max}[\rightarrow H$ of $(5.59)$ with either $T\_{\max}=+\infty$, or $T\_{\max}<\infty$ and $\lim\_{t \uparrow T\_{\max}}\|u(t)\|=+\infty$. If $u\_0 \in D(A)$, the mild solution $u$ is classical. (2) If $F$ is Lispschitz continuous, then $T\_{max}=+\infty.$
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My question:



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> I would like to know if **Theorem 5.8** is valid for $ A $ being just the infinitesimal generator of a $C\_0$-semigroup, that is, **without** the hypothesis that $\|S(t)\|\leq 1$.
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I imagine that the answer is positive in the case that $S(t)$ is uniformly bounded, that is, there is $M>0$ such that $\|S(t)\|\leq M$ for $t \geq 0$. 


Assuming I have proven that $A-\lambda I$ is the infinitesimal generator of a $C\_0$-semigroup of contractions, for some $\lambda>0$. Then, $A$ is the infinitesimal generator of a $C\_0$-semigroup $T(t)$ satisfyng $\|T(t)\|\leq e^{\lambda t}$, for all $t \geq 0$. I don't know if the result is valid in this case.


Can anyone help me? Please.



$u_t=Au+F(u)$ where $A$ is the infinitesimal generator of $C_0$-semigroup
Yes the theorem is true in the general case. The semigroup is assumed to be contractive for the sake of simplicity. Of course the assumption is not restrictive since there is always an equivalent norm which makes the semigroup contractive (this is the same idea in the proof of **Hille-Yosida theorem** to pass from the uniformly bounded case to the contractive case). You can find the general statement 
(for Lipschitz continuous functions) of the theorem in **Theorem 1.4 pp 185.** (see also Theorem 1.2) of the book



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> A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer 1983.
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**Edit:**
As for the second part on the **improved regularity**, the result is true for locally Lipschitz functions, when $X$ is **reflexive** (in your case it is a Hilbert space). See Proposition 4.3.9 pp 60 (and Remark 4.3.10) in the book



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> T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford 1998.
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**PS:** One should note the definition of Lipschitz continuous function in pp 55. 


