Return to Question

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?

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?

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

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?