The heat operator, the heat equation, and the Cauchy problem

Let $n\ge 1$ and let $\mathrm{\Omega }\subseteq {\mathbb{R}}^{n+1}$ be open in the space-time variable $(x,t)\in {\mathbb{R}}^n\times \mathbb{R}$; points of $\mathrm{\Omega }$ are thus written with a spatial slot $x\in {\mathbb{R}}^n$ and a time slot $t\in \mathbb{R}$. Partial derivatives are those of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$, multi-indices and the classes $C^k$ are those of $C^k$ maps and multi-index derivative notation in Euclidean space, and ${\mathrm{\Delta }}_x$ denotes the Laplacian of The Laplacian of a $C^2$ function and of a $C^2$ vector field applied in the spatial variables with $t$ held fixed. The vocabulary of differential operators, their order, and of classical solutions is that of Scalar partial differential equations, order, and classical solutions.

The heat operator is ${\mathrm{\partial }}_t-{\mathrm{\Delta }}_x$, a linear second-order operator in the sense of Linear, semilinear, quasilinear, and fully nonlinear partial differential equations. A classical solution of the heat equation on $\mathrm{\Omega }$ is a function $u\in C^2(\mathrm{\Omega })$ with

${\mathrm{\partial }}_{t}u-{\mathrm{\Delta }}_{x}u=0\text{on }\mathrm{\Omega },$

and the inhomogeneous heat equation is the equation ${\mathrm{\partial }}_{t}u-{\mathrm{\Delta }}_{x}u=f$ for a prescribed source $f$. A complex-valued $u$ is a classical solution when its real and imaginary parts are. Since the spatial quadratic form of ${\mathrm{\partial }}_t-{\mathrm{\Delta }}_x$ is positive definite while the time direction enters only through a first derivative, the operator is parabolic at every point in the classification of Elliptic, hyperbolic, and parabolic principal symbols.

A Cauchy problem for the heat equation on a space-time domain consists of the equation together with initial data $u(\cdot ,0)=u_0$ prescribed on the time-zero slice; values prescribed on the lateral (spatial) boundary of the domain are boundary data. Initial and spatial boundary data are different sets of constraints and are not interchanged.