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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?

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    $\begingroup$ Is your question about the norm $\| x(t)\|$ being a space-time norm? Or do you mean point-wise in time? Since you tagged parabolic pde, if you just think of the heat equation on a periodic domain, the energy can dissipate arbitrarily fast, so that puts some constraints on what kinds of statements you can make. Can you be a bit more detailed in your question? $\endgroup$ – Willie Wong Dec 13 '19 at 14:51
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    $\begingroup$ Thank you for your comment. I edited the post. I hope it's a bit clear now. $\endgroup$ – Sigma Dec 13 '19 at 18:09
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    $\begingroup$ An idea used in interpolation theory is to define a norm on functions of x by considering them as initial data for a heat equation and imposing a norm on the corresponding solution. This seems pretty close to your idea $\endgroup$ – Piero D'Ancona Dec 16 '19 at 0:26
  • $\begingroup$ This is an interesting question. Maybe one have to start from a very simple example! $\endgroup$ – S. Maths Dec 16 '19 at 0:27
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    $\begingroup$ Triebel Function spaces II is a good starting point. The keyword is 'thermic' characterization of function spaces $\endgroup$ – Piero D'Ancona Dec 17 '19 at 8:07
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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 $t>0$.

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  • $\begingroup$ Thank you Sir. Do you have any idea to overcome the loss of information? $\endgroup$ – Sigma Dec 19 '19 at 18:50
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    $\begingroup$ @Sigma . I'll think about it. Meanwhile, if you agree that this answers your question, you are welcome to accept it, so that I can be granted the bounty :) $\endgroup$ – Denis Serre Dec 20 '19 at 10:41
  • $\begingroup$ Sure. Thank you! $\endgroup$ – Sigma Dec 20 '19 at 10:55

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