Caloric function

Remarks

Example: on $\R \times \R$, the function $u(x, t) = x^2 + 2t$ is caloric, since $\partial_t u = 2$ and $\partial_x^2 u = 2$. On $\R^n \times (0, \infty)$, the heat kernel $\Phi(x, t) = (4 \pi t)^{-n/2} e^{-\abs{x}^2 / (4t)}$ is caloric.

Non-example: $u(x, t) = x^2 + t$ on $\R \times \R$ is smooth but not caloric, because $\partial_t u = 1$ while $\partial_x^2 u = 2$.

Some authors write the heat equation as $\partial_t u = k \Delta u$ with a constant $k > 0$; a change of the time scale reduces this to $k = 1$. A function with $\partial_t u \le \Delta u$ is called subcaloric, and one with $\partial_t u \ge \Delta u$ is called supercaloric. Caloric functions are also called temperatures.