Unit 5: The Heat/Diffusion Equation: Maximum Principles, Fundamental Solution & Separation of Variables

1. Thermodynamic Conservation of Thermal Energy

Consider an arbitrary, bounded, smooth spatial domain $V \subset \mathbb{R}^3$ filled with a rigid isotropic conducting material of mass density $\rho(\mathbf{x})$, specific heat capacity $c_p(\mathbf{x})$, and thermal conductivity $k(\mathbf{x})$. Let $u(\mathbf{x}, t)$ denote the absolute temperature field at position $\mathbf{x} = (x, y, z)$ and time $t$.

The total thermal energy contained in the volume $V$ is:

By the First Law of Thermodynamics, the rate of increase of internal thermal energy equals the net rate of heat conducted inward across the boundary surface $\partial V$ plus the rate of heat generated by internal volumetric sources $Q(\mathbf{x}, t)$:

where $\mathbf{q}(\mathbf{x}, t)$ is the heat flux vector (energy per unit area per unit time) and $\mathbf{n}$ is the outward unit normal.

Applying the Divergence Theorem to the surface flux integral:

Combining all terms under a single volume integral:

Since this integral vanishes for every arbitrary subvolume $V \subset \mathbb{R}^3$, the integrand must vanish identically, yielding the continuity equation for heat energy:

2. Fourier's Law of Heat Conduction

In 1822, Joseph Fourier established the empirical constitutive law: heat flows in the direction of the steepest decrease in temperature at a rate proportional to the temperature gradient:

where $k > 0$ is the thermal conductivity of the medium. Substituting Fourier's law into the continuity equation:

For a homogeneous, isotropic medium where $\rho, c_p,$ and $k$ are uniform constants:

Defining the thermal diffusivity $\alpha = \frac{k}{\rho c_p} > 0$ (with physical dimensions $[L^2 T^{-1}]$), we obtain in the absence of internal sources ($Q = 0$):

In one spatial dimension ($x \in \mathbb{R}$), this simplifies to the 1D heat equation:

1. The Parabolic Cylinder and Maximum Principle

Let $\Omega \subset \mathbb{R}^n$ be an open, bounded spatial domain, and let $T > 0$ be a fixed time horizon. Define the spacetime cylinder:

and the parabolic boundary $\Gamma_T$ consisting of the base at $t = 0$ and the lateral spatial boundary:

Notice that the top lid $\Omega \times \{t = T\}$ is part of $Q_T$, not $\Gamma_T$.

Theorem 5.1 (Weak Maximum Principle for the Heat Equation): Let $u \in C^2(Q_T) \cap C(\bar{Q}_T)$ satisfy the heat equation:

Then the maximum and minimum of $u$ on the closed cylinder $\bar{Q}_T$ are attained on the parabolic boundary $\Gamma_T$:

Proof: Let $M = \max_{\Gamma_T} u$. We must show that $u(\mathbf{x}, t) \le M$ for all $(\mathbf{x}, t) \in Q_T$. Assume for contradiction that there exists a point $(\mathbf{x}_0, t_0) \in Q_T$ such that $u(\mathbf{x}_0, t_0) = M + \epsilon$ with $\epsilon > 0$. For any $\delta > 0$, construct the auxiliary perturbation function:

Choose $\delta = \frac{\epsilon}{2T} > 0$. Then on $\Gamma_T$:

While at $(\mathbf{x}_0, t_0)$:

Since $\bar{Q}_T$ is compact and $v$ is continuous, $v$ must attain its global maximum on $\bar{Q}_T$ at some point $(\mathbf{x}^, t^) \in Q_T$ (in the interior or on the top lid $t^* = T$). At this internal maximum point:

1. Spatial first derivatives vanish: $\nabla v(\mathbf{x}^, t^) = \mathbf{0}$.
2. Spatial second derivatives satisfy the negative semi-definite Hessian condition: $\nabla^2 v(\mathbf{x}^, t^) \le 0$.
3. The time derivative must satisfy: $v_t(\mathbf{x}^, t^) \ge 0$ (if $t^ < T$, $v_t = 0$; if $t^ = T$, $v_t \ge 0$ from below).

Now compute the heat operator applied to $v$:

However, using the derivative conditions at the maximum point:

This yields the strict contradiction:

Therefore, no such point $(\mathbf{x}_0, t_0)$ can exist, proving $u(\mathbf{x}, t) \le M$ throughout $\bar{Q}_T$. Applying the same argument to $-u$ yields the minimum principle. $\blacksquare$

2. Uniqueness and Continuous Dependence

Theorem 5.2 (Uniqueness of Initial-Boundary Value Problem): There exists at most one classical solution to the Dirichlet problem:

Proof: Let $u_1$ and $u_2$ be two solutions, and let $w = u_1 - u_2$. Then $w_t - \alpha \nabla^2 w = 0$ in $Q_T$, and $w = 0$ on $\Gamma_T$. By the Weak Maximum and Minimum Principles:

Thus $w(\mathbf{x}, t) \equiv 0$, which proves $u_1 \equiv u_2$. $\blacksquare$

1. Homogeneous Dirichlet Conditions on a Finite Rod $[0, L]$

Consider the heat conduction on $[0, L]$ with both ends kept at zero temperature:

Separating variables $u(x, t) = X(x) T(t)$:

1. Spatial Sturm-Liouville problem: $X'' + \lambda X = 0, X(0) = X(L) = 0$.

Eigenvalues: $\lambda_n = \left(\frac{n\pi}{L}\right)^2$, Eigenfunctions: $X_n(x) = \sin\left(\frac{n\pi x}{L}\right)$ for $n \in \mathbb{N}$.

1. Temporal ODE: $T_n'(t) + \alpha \lambda_n T_n(t) = 0 \implies T_n(t) = e^{-\alpha (n\pi/L)^2 t}$.

General solution:

where the Fourier sine coefficients are:

Notice the exponential decay factor: higher Fourier harmonics damp out drastically faster ($n^2$), smoothing out initial irregularities instantly!

2. Homogeneous Neumann Conditions (Insulated Rod)

When both ends of the rod are perfectly thermally insulated:

The spatial eigenfunctions are $X_0(x) = 1$ (for $\lambda_0 = 0$) and $X_n(x) = \cos\left(\frac{n\pi x}{L}\right)$. The general solution is:

where:

As $t \to \infty$, all decaying exponential terms vanish:

The temperature asymptotically equilibrates to the exact average initial temperature of the rod, demonstrating total thermal energy conservation in an insulated system.

1. Inhomogeneous Initial-Boundary Value Problems

Consider the heat equation with an external heat source $f(x, t)$:

2. Duhamel's Principle

Jean-Marie Duhamel (1837) proved that the solution to an inhomogeneous PDE driven by a continuous source $f(x, t)$ can be constructed by integrating a family of solutions to homogeneous initial value problems.

Theorem 5.3 (Duhamel's Principle for Parabolic Equations): For each fixed parameter $\tau \ge 0$, let $w(x, t; \tau)$ be the unique solution to the homogeneous heat problem for $t > \tau$:

Then the solution $u(x, t)$ to the inhomogeneous problem with zero initial condition is given by the integral:

Proof: Differentiating $u(x, t)$ with respect to $t$ via Leibniz's rule for differentiating under the integral sign:

By the initial condition of the family $w$, $w(x, t; t) = f(x, t)$. Next, compute the spatial derivative:

Now evaluate the heat operator on $u$:

Since $w$ satisfies the homogeneous heat equation $w_t - \alpha w_{xx} = 0$, the integral vanishes identically:

Furthermore, at $t = 0$:

And along the boundaries $x = 0, L$, since $w(0, t; \tau) = w(L, t; \tau) = 0$, $u(0, t) = u(L, t) = 0$. $\blacksquare$

1. Derivation of the Heat Kernel via Similarity Variables

Consider the heat equation on the unbounded line $\mathbb{R}$:

subject to a unit point source concentrated at the origin at $t = 0$:

where $\delta(x)$ is the Dirac delta distribution.

The PDE and total heat content $\int_{-\infty}^\infty u(x, t)\,dx = 1$ are invariant under the parabolic scaling transformation:

This scaling symmetry dictates that $u(x, t)$ must take the self-similar form:

Compute partial derivatives:

Substituting into $u_t = \alpha u_{xx}$:

Integrating once with respect to $\xi$:

Since $u \to 0$ and $u_x \to 0$ as $|x| \to \infty$, the constant $C = 0$:

Substituting $\xi = \frac{x}{\sqrt{\alpha t}}$:

To determine the normalization constant $A$, enforce $\int_{-\infty}^\infty u(x, t)\,dx = 1$:

Definition 5.1 (The Gaussian Heat Kernel): The fundamental solution (or heat kernel) of the 1D diffusion equation is:

For an arbitrary bounded initial temperature distribution $u(x, 0) = f(x)$, the solution is given by the convolution: