The Heat Equation (or diffusion equation) is the prototypical parabolic partial differential equation. It models processes where a quantity spreads from regions of high concentration to low concentration, smoothing irregularities as time evolves.

[motivation] ## Motivation ### Physical Origins If $u(x, t)$ represents the temperature at position $x$ and time $t$, Fourier's law states that the heat flux is proportional to the negative temperature gradient: $q = -k \nabla u$. Conservation of energy requires $\partial_t u + \nabla \cdot q = 0$. Combining these gives $\partial_t u = k \Delta u$, which (after normalising $k = 1$) is the heat equation. Beyond thermodynamics, the same equation governs Brownian motion, chemical diffusion, and — through the connection to [semigroup theory](/page/Semigroup%20Theory) — serves as the foundation for parabolic PDE theory. ### Why Not Just Solve It Pointwise? For the heat equation on all of $\mathbb{R}^n$, one might attempt a direct [Fourier transform](/page/Fourier%20Transform) approach: $\hat{u}_t = -|\xi|^2 \hat{u}$ gives $\hat{u}(\xi, t) = \hat{g}(\xi) e^{-|\xi|^2 t}$, and inverting produces a convolution $u = \Phi(\cdot, t) * g$. This works well for the Cauchy problem with smooth, rapidly decaying data — but it says nothing about bounded domains, [boundary](/page/Boundary) conditions, maximum principles, or the qualitative structure of solutions. ### The Role of Symmetry and Mean Values The deeper approach, developed in this article, exploits the *structure* of the equation rather than explicit formulas. The heat equation has a parabolic scaling symmetry — invariance under $(x, t) \mapsto (\lambda x, \lambda^2 t)$ — which determines the form of the fundamental solution via a self-similar ansatz. Once the fundamental solution is in hand, a mean value formula (analogous to the elliptic mean value property) unlocks the qualitative theory: maximum principles, uniqueness, regularity, and derivative estimates all follow from it. [/motivation]

## Formal Definition

[definition: Heat Equation] The homogeneous **Heat Equation** for a function $u: \Omega \times (0, \infty) \to \mathbb{R}$ is the second-order linear PDE: \begin{align*} \partial_t u - \Delta u = 0, \end{align*} where $\Omega \subseteq \mathbb{R}^n$ is an [open set](/page/Open%20Set), $t > 0$ denotes time, and $\Delta = \sum_{i=1}^n \partial_{x_i}^2$ is the spatial Laplacian. [/definition]

## Derivation of the Fundamental Solution

Our first goal is to find a specific solution on $\mathbb{R}^n \times (0, \infty)$ by exploiting the scaling symmetry of the equation.

### Scaling Invariance

If $u(x, t)$ solves the heat equation and $\lambda > 0$, the rescaled function $u_\lambda(x, t) := u(\lambda x, \lambda^2 t)$ also solves it, since both $\partial_t$ and $\Delta_x$ pick up a factor of $\lambda^2$ under this substitution. This invariance under the dilation $(x, t) \mapsto (\lambda x, \lambda^2 t)$ suggests looking for self-similar solutions of the form

\begin{align*} u(x, t) = \frac{1}{t^{n/2}} v\!\left(\frac{x}{\sqrt{t}}\right), \end{align*}

where the exponent $n/2$ is chosen to ensure conservation of total mass: $\int_{\mathbb{R}^n} u \, d\mathcal{L}^n$ is independent of $t$.

### Reduction to an ODE

Substituting the ansatz $u(x, t) = t^{-n/2} v(y)$ with $y = x/\sqrt{t}$ into $\partial_t u = \Delta u$ and cancelling the common factor $t^{-(n+2)/2}$ yields the elliptic equation

\begin{align*} \Delta v + \frac{1}{2} y \cdot \nabla v + \frac{n}{2} v = 0. \end{align*}

Assuming radial symmetry $v(y) = w(|y|)$ and writing $r = |y|$, the radial Laplacian $\Delta v = w'' + \frac{n-1}{r} w'$ reduces this to

\begin{align*} w'' + \left(\frac{n-1}{r} + \frac{r}{2}\right) w' + \frac{n}{2} w = 0. \end{align*}

Multiplying by $r^{n-1}$ reveals a total derivative: $(r^{n-1} w')' + \frac{1}{2}(r^n w)' = 0$. Integrating once and imposing decay at infinity gives $w'/w = -r/2$, hence $w(r) = b \, e^{-r^2/4}$. The normalisation $\int_{\mathbb{R}^n} u \, d\mathcal{L}^n = 1$ fixes $b = (4\pi)^{-n/2}$.

[definition: Fundamental Solution] The **fundamental solution** (or heat kernel) of the heat equation is the function $\Phi: \mathbb{R}^n \times (\mathbb{R} \setminus \{0\}) \to \mathbb{R}$ defined by: \begin{align*} \Phi(x, t) := \begin{cases} \dfrac{1}{(4\pi t)^{n/2}} \, e^{-|x|^2/(4t)} & \text{for } x \in \mathbb{R}^n, \, t > 0, \\ 0 & \text{for } x \in \mathbb{R}^n, \, t < 0. \end{cases} \end{align*} [/definition]

The fundamental solution is smooth, satisfies the heat equation, has unit mass for each $t > 0$, and converges to $\delta_0$ as $t \downarrow 0$.

[quotetheorem:53]

The delta-[limit](/page/Limit) property (point 4) is the key to the initial value problem: it means that convolving $\Phi(\cdot, t)$ with initial data $g$ should recover $g$ as $t \to 0$.