Consider the heat equation in $\mathbb{R}^2$. Let $u=u(t, x)$ is s solutio $\mathrm{n}$ to
$$
\left\{\begin{array}{l}
\frac{\partial u}{\partial t}-\Delta u=0 \\
\left.u\right|_{t=0}=u_0 \in L^2
\end{array}\right.
$$
Then there exists a universal constant $C$ such that
$$
\int_0^{\infty}\|u(t)\|_{L^2}^2 \mathrm{~d} t \leq C\left\|u_0\right\|_{L^2}^2
$$