# Inequality for initial data

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) $$\|x_0\|_X\le C_1\|x(t)\|_X+C_2\|x(\tau)\|_X+C_3 \|x'(t)\|_X, \qquad 0 for a fixed $$\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?

• 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? – Willie Wong Dec 13 '19 at 14:51
• Thank you for your comment. I edited the post. I hope it's a bit clear now. – Sigma Dec 13 '19 at 18:09
• 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 – Piero D'Ancona Dec 16 '19 at 0:26
• This is an interesting question. Maybe one have to start from a very simple example! – S. Maths Dec 16 '19 at 0:27
• Triebel Function spaces II is a good starting point. The keyword is 'thermic' characterization of function spaces – Piero D'Ancona Dec 17 '19 at 8:07

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 $$t>0$$.