Suppose \(U\subseteq \mathbb{R}^d\) is open. The heat equation is
\[ u_t - \Delta u = 0 \quad \text{in } U \times (0, \infty). \]This equation is also called homogeneous heat equation. The inhomogeneous heat equation is given by
\[ u_t - \Delta u = f \quad \text{in } U \times (0, \infty). \]for a suitable \(f\colon U\times (0,\infty )\to \mathbb{R}\).
Remarks
Special solutions Link to heading
Existence and Uniqueness Link to heading
- On bounded domains there is at most one solution.
- For unbounded domains there is more than one solution (but only one physically meaningful) (see Cauchy problem ).