The Heat Equation - Key Properties

The heat equation exhibits remarkable smoothing and decay properties that reflect the dissipative nature of diffusion processes.

DefinitionMaximum Principle

Strong Maximum Principle: If $u$ satisfies $u_t = k\nabla^2 u$ in a space-time domain $\Omega \times (0, T]$ , then:

The maximum of $u$ occurs either at $t = 0$ or on the spatial boundary $\partial\Omega$
If the maximum occurs at an interior point $(x_0, t_0)$ with $t_0 > 0$ , then $u$ is constant in $\Omega \times [0, t_0]$

This principle has immediate physical interpretation: temperature cannot develop hot spots in the interior—the hottest point is either present initially or maintained by boundary conditions.

Infinite Propagation Speed: Unlike wave propagation, heat diffusion has infinite speed. If $u(x,0)$ has compact support, then $u(x,t) > 0$ for all $x$ and any $t > 0$ . However, the effect decays exponentially with distance.

DefinitionSmoothing Property

Solutions to the heat equation are infinitely differentiable for all $t > 0$ , regardless of the regularity of initial data. If $u(x,0) = f(x)$ with $f$ merely continuous (or even in $L^2$ ), then $u(x,t) \in C^\infty$ for all $t > 0$ .

Remark

This smoothing property contrasts sharply with the wave equation, which preserves the regularity of initial data. The heat equation acts as a "mollifier," immediately smoothing out discontinuities and irregularities.

ExampleEnergy Decay

For the heat equation on a bounded domain with homogeneous boundary conditions, the $L^2$ energy: $E(t) = \int_\Omega u^2(x,t)\,dx$ satisfies: $\frac{dE}{dt} = -2k\int_\Omega |\nabla u|^2\,dx \leq 0$

Energy monotonically decreases, approaching zero as $t \to \infty$ . This reflects the irreversible approach to thermal equilibrium.

Uniqueness: The maximum principle immediately implies uniqueness. If $u_1$ and $u_2$ are two solutions with the same initial and boundary data, then $w = u_1 - u_2$ satisfies $w_t = k\nabla^2 w$ with zero initial and boundary data. By the maximum principle, $w \equiv 0$ .

These properties make the heat equation fundamentally different from elliptic and hyperbolic PDEs, reflecting the physics of irreversible diffusion processes.