I'm wondering if there is any idea to bound the norm of the initial data by the norm of corresponding solution? To make it clear, consider the following abstract Cauchy problem: $x'(t)=Ax(t)$, in $(0,T)$, with $x(0)=x\_0$, where $A$ generates an **analytic** $C\_0$-semigroup on a Banach space $X$. I'm looking for some ideas to get an inequality of type (or something similar) $$\|x\_0\|\_X\le C\_1\|x(t)\|\_X+C\_2\|x(\tau)\|\_X+C\_3 \|x'(t)\|\_X, \qquad 
0<t<T,$$ for a fixed $\tau$ and positive constants $C\_i$, for all initial data in a set $A=\{x\_0 \in D\colon \|x\_0\|\_1 \le M\}$, for some constant $M$, a domain $D$ and a norm $\|\cdot\|\_1$.


If for example $X$ is a Hilbert space, and the operator $A$ is self-adjoint dissipative, we already have $\|x(t)\|\le \|x\_0\|$ for all $t$. So I'm looking for restrictions on initial data that yields the desired inequality. 


I guess a result from regularity theory (for analytic semigroups for example) or something similar would help. Do you know any similar idea? 



Inequality for initial data
I fear that the question is posed in too general terms. As stated, the answer is **No**, because of the following example.


Consider the Hilbert space $X=L^2(0,1)$ and the operator $S\_t\in{\cal L}(X)$ defined by
$$(S\_ta)(x)=\left\{\begin{array}{lr}
0, & x\in(0,t), \\ a(x-t), & x\in(t,1).
\end{array}\right.$$
This defines a semi-group over $X$, which corresponds to the initial-boundary-value problem
$$\partial\_tu+\partial\_xu=0,\qquad u(0,x)=a(x),\qquad u(t,0)=0.$$
Observe that $S\_ta\equiv0$ for $t\ge1$. Therefore you cannot estimates the initial data from $u(t)$ for $t\ge1$. Actually, because you do loose information even at time $t\in(0,1)$, you cannot estimate $u(0)$ from $u(t)$, whenever $t>0$.


