Suppose \(g\in C(\mathbb{R}^d)\cap L^\infty (\mathbb{R}^d)\) at \(t=0\) and \(\phi\) is the fundamental solution of the heat equation . The convolution

\[ u(x,t) = \int_{\mathbb{R}^d} \varphi(x - y, t) g(y) \, dy \]

solves the Cauchy problem

\[ u_t - \Delta u = 0 \quad \text{in } \mathbb{R}^d \times (0, \infty) \]

with \(u=g\) on \(\mathbb{R}^d\times \{0\}\).

To be more precise the following properties are satisfied

  • \(u \in C^\infty(\mathbb{R}^n \times (0,\infty))\) (because \(\varphi\) is smooth and by some integration theory),
  • \(u_t - \Delta u = 0\) and
  • \(u(x,t) \to g(x_0)\) as \((x,t) \to (x_0, 0)\) for all \(x_0 \in \mathbb{R}^d\).
Remarks
  • The above given \(u\) has infinite propagation of speed, i.e. if \(g \geq 0\) and \(g \not\equiv 0\), then \(u(x,t)>0\) for all \(t>0\) and \(x\in \mathbb{R}^d\).
  • If \(g(x) \in [a, b]\), then \(u(x,t) \in [a, b]\) for every \(t>0\).
  • Suppose \(g \in L^1(\mathbb{R}^d)\). Then \(u(\cdot,t) \in L^1(\mathbb{R}^d)\) with \[ \int_{\mathbb{R}^d} u(x,t)\,dx = \int_{\mathbb{R}^d} g(x)\,dx \] for every \(t>0\). We say the total mass is conserved.
  • Suppose \(p \in [1, \infty]\) and \(g \in L^p(\mathbb{R}^d)\). Then \(u \in C^\infty(\mathbb{R}^d \times (0,\infty))\) and it solves the heat equation with \[ \| u(\cdot,t) - g \|_{L^p(\mathbb{R}^d)} \to 0 \quad \text{as} \quad t \to 0. \]