The Cauchy problem is the heat equation with some initial condition \(g\in C(\mathbb{R}^d)\cap L^\infty (\mathbb{R}^d)\) at \(t=0\), i.e.
\[ u_t - \Delta u = 0 \quad \text{in } \mathbb{R}^d \times (0, \infty) \]with \(u=g\) on \(\mathbb{R}^d\times \{0\}\).
Remarks