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 $t$. However, we can estimate $\|x\|_{W^{1,p}(0,T;X)}$ and use Sobolev embedding $W^{s,p}(0,T;X) \subset C([0,T];X)$ to estimate $\|x_0\|_X$ under restrictions on $p$ and $s$. 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?