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If we have some $C_0$-semigroup $(S(t))_{t\geq 0}$ and $\lVert u(t)\rVert\leq C$ for $t\in [0,T)$ and $u(0)=u_0$, why do we then have, for $0<t<t'<T$, the estimation $$ \lVert (S(t)-S(t'))u_0\rVert\leqslant C\lvert t-t'\rvert? $$

This should be an easy consequence of the fact that $(S(t))$ is a $C_0$-semigroup but I do not see the reason.

Pedro
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Rhjg
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  • Do you mean that it follows from the axioms of the $C^0$-semigroup? – JJR May 19 '17 at 15:46
  • This would be my conjecture. At least in the script there is the short remark: "$C_0$-semigroup: $t,t'>0$" to this estimate. – Rhjg May 19 '17 at 15:48
  • Do you know anything about $u$? – JJR May 19 '17 at 15:56
  • I guess that $u$ is a mild solution of a PDE $u_t=Au+F(u), u(0)=u_0$ where $S(t)$ is the solution Operator to $u_t=Au$, i.e. $u(t)=S(t)u_0+\int_{0}^t S(t-s)F(u(s)), ds$. – Rhjg May 19 '17 at 16:02

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