44 2. FOUR IMPORTANT LINEAR PDE 2.3. HEAT EQUATION Next we study the heat equation (1) ut − Δu = 0 and the nonhomogeneous heat equation (2) ut − Δu = f, subject to appropriate initial and boundary conditions. Here t 0 and x ∈ U, where U ⊂ Rn is open. The unknown is u : ¯ U × [0, ∞) → R, u = u(x, t), and the Laplacian Δ is taken with respect to the spatial variables x = (x1,...,xn): Δu = Δxu = ∑n i=1 uxixi . In (2) the function f : U×[0, ∞) → R is given. A guiding principle is that any assertion about harmonic functions yields an analogous (but more complicated) statement about solutions of the heat equation. Accordingly our development will largely parallel the correspond- ing theory for Laplace’s equation. Physical interpretation. The heat equation, also known as the diffusion equation, describes in typical applications the evolution in time of the density u of some quantity such as heat, chemical concentration, etc. If V ⊂ U is any smooth subregion, the rate of change of the total quantity within V equals the negative of the net flux through ∂V : d dt V u dx = − ∂V F · ν dS, F being the flux density. Thus (3) ut = − div F, as V was arbitrary. In many situations F is proportional to the gradient of u but points in the opposite direction (since the flow is from regions of higher to lower concentration): F = −aDu (a 0). Substituting into (3), we obtain the PDE ut = a div(Du) = aΔu, which for a = 1 is the heat equation. The heat equation appears as well in the study of Brownian motion.

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