For \(d\ge 1\) the function
\[ \phi(x,t) = \frac{1}{(4\pi t)^{d/2}} e^{-\frac{|x|^2}{4t}} \cdot 𝟙_{(0,\infty)}(t) \]is called fundamental solution of the heat equation .
Remarks
- For \(t>0\) the fundamental solution solves the heat equation.
- The factor in the definition of \(\phi\) ensures a normalization property, i.e. \[ \int_{\mathbb{R}^d} \varphi(\cdot, t) = 1 \] for every \(t>0\).
- The fundamental solution gives a solution of the Cauchy problem .
- The fundamental solution satisfies a semi group property, i.e. for \(t, s > 0\), \[ \varphi(\cdot, t) * \varphi(\cdot, s) = \varphi(\cdot,\,t + s). \]