Heat Conduction and Diffusion

The heat equation $u_t = \alpha\nabla^2 u$ models diffusion of heat, probability, and chemical concentration. Its solutions exhibit smoothing (instantaneous regularization), decay to equilibrium, and maximum principles fundamentally different from wave propagation.

The Heat Kernel

Definition6.5Fundamental Solution of the Heat Equation

The heat kernel (fundamental solution) on $\mathbb{R}^n$ is $K(\mathbf{x},t) = \frac{1}{(4\pi\alpha t)^{n/2}}\exp\!\left(-\frac{|\mathbf{x}|^2}{4\alpha t}\right)$ for $t > 0$ . The solution to the initial value problem $u_t = \alpha\nabla^2 u$ , $u(\mathbf{x},0) = f(\mathbf{x})$ is $u(\mathbf{x},t) = \int_{\mathbb{R}^n} K(\mathbf{x}-\mathbf{y},t)f(\mathbf{y})\,d^n y = (K_t * f)(\mathbf{x})$ . The heat kernel is a Gaussian that broadens as $\sqrt{t}$ , reflecting diffusive spreading.

ExampleHeat Equation on a Finite Interval

On $[0,L]$ with $u(0,t) = u(L,t) = 0$ and $u(x,0) = f(x)$ : $u(x,t) = \sum_{n=1}^\infty b_n \sin(n\pi x/L)e^{-\alpha n^2\pi^2 t/L^2}$ where $b_n = \frac{2}{L}\int_0^L f(x)\sin(n\pi x/L)\,dx$ . Each mode decays exponentially; higher modes decay faster ( $\sim e^{-n^2 t}$ ), producing the smoothing effect. For long times: $u \approx b_1\sin(\pi x/L)e^{-\alpha\pi^2 t/L^2}$ .

Properties

Definition6.6Maximum Principle for the Heat Equation

(Weak maximum principle) A solution $u$ of $u_t \leq \alpha\nabla^2 u$ on $\Omega \times (0,T]$ attains its maximum on the parabolic boundary $\partial_p(\Omega\times(0,T]) = (\Omega\times\{0\}) \cup (\partial\Omega\times[0,T])$ . This means: the temperature cannot exceed its initial and boundary maximum without a heat source. The strong maximum principle further states that if the maximum is attained at an interior point, $u$ is constant.

RemarkInfinite Speed of Propagation

Unlike the wave equation, the heat equation has infinite speed of propagation: if $f$ has compact support, then $u(x,t) > 0$ for all $x$ and all $t > 0$ (since $K > 0$ everywhere). A point source at $t = 0$ is instantly felt everywhere, though the effect is exponentially small at large distances. This is physically reasonable for diffusion (molecular motion reaches everywhere) but distinguishes parabolic from hyperbolic equations fundamentally.