Theorem: Let $X$ be a Banach space, $\{T(t)\}_{t\geq 0}$ a $C_0$-semigroup on $X$ and $U_0\in D(A)$. If $A:D(A)\subset X\to X$ is the infinitesimal generator of $\{T(t)\}_{t\geq0}$, then the function $U:[0,\infty)\to X$ given by $U(t)=T(t)U_0$ is a solution of $(1)$. $$\left\{\begin{align*} U_t(t)=AU(t);&~~~~t\in[0,\infty)\\ U(0)=U_0& \end{align*}\right.\tag{1}$$
Now consider the problem
$$\left\{\begin{align*} y_t(x,t)=y_{xx}(x,t);&~~~~&&x\in\mathbb{R};\;t\in[0,\infty)\\ ~y(x,0)=f(x);&&&x\in\mathbb{R} \end{align*}\right.\tag{2}$$
where $y_{xx}$ is the weak derivative of second order of $y$. By theorem above, it's possible to show that $(2)$ has a solution.
So, could someone explain me (with some details) how can we rewrite $(2)$ in order to get a equivalent system, analogous to $(1)$?
The solution that I saw just says that it's enough to show that the operator $A:H^2(\mathbb{R})\to L^2(\mathbb{R})$ defined by $A(y)=y_{xx}$ is a infinitesimal generator of a $C_0$-semigroup on $L^2(\mathbb{R})$ (for this, the Hille-Yosida theorem is used however my question is not about the application of the Hille-Yosida Theorem. I need help to understand how to transform the original system in a system like $(1)$ and why the existence of a solution for $(1)$ implies the exitece of a solution for the original system).
Thanks.
It's pretty straightforward. Some of it is already in the solution you cited. Here $X=L_2(\mathbb R)$, $A=\partial^2_x$, $D(A)=H^2(\mathbb R)$, $U_0$=$f$. Applying the theorem one gets the result for (2).