We study the heat equation for one spatial dimension on the real line with some initial datum: $\begin{align} \partial_{t}u &= \partial_{xx}u(x,t), & (x,t)\in \mathbb{R}\times (0,\infty) \\ u(x,0) &= \phi(x), & x\in \mathbb{R} \end{align}$where $\phi\in C^1(\mathbb{R})$. # Properties **Comparison to heat equation:** - If the initial condition has a compact support, then the solution of the **wave equation** has a compact support at all times. However, it is not true that compact support of the initial condition for the **heat equation** implies compact support of the solution at later times. In fact, unless the initial condition is identically zero, the solution becomes immediately positive everywhere. This is because the heat equation on $\mathbb{R}^n$ admits an explicit solution given by convolution with a Gaussian kernel, which is strictly positive for all $x$ and $t > 0$. As a result, any nonzero compactly supported initial data leads to a solution that is nonzero on all of $\mathbb{R}^n$ for any $t > 0$. This reflects the infinite speed of propagation characteristic of the heat equation: thermal disturbances spread instantaneously across the entire domain, unlike in wave propagation, where signals travel at finite speed.