6 The Heat Equation

Fourier’s law says that heat flows from hot to cold at a rate proportional to the temperature gradient, \(q = -K\nabla u\). Combined with conservation of energy, \(\rho c_pu_t + \operatorname {div}q = 0\), it gives the heat equation \begin {equation} \label {eq:heat} u_t = \kappa \Delta u, \qquad \kappa = \frac {K}{\rho c_p} , \end {equation} with diffusivity \(\kappa \). The same equation governs the concentration of a diffusing substance (Fick’s law) and the probability density of a Brownian particle (Einstein). Where the wave equation transports and conserves, the heat equation smooths and dissipates. This section constructs its fundamental solution, proves the maximum principle, and draws the consequences: smoothing, infinite speed of propagation, uniqueness and stability.

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