Heat equation - Knowlpedia

The heat equation with diffusivity $\nu>0$ is

\partial_tu=\nu\Delta u.

Here $u=u(x,t)$ , and the Laplacian acts on the spatial variables. A classical solution has the derivatives required by this pointwise equality. A prescribed source gives the inhomogeneous equation $\partial_tu-\nu\Delta u=f$ .

Data and interpretation

An evolution problem also specifies initial data and, on a domain with boundary, boundary conditions. The positive sign of $\nu$ makes Fourier modes decay forward in time. A component of a vector field expressed in a moving coordinate basis need not satisfy the scalar heat equation: differentiating the basis can introduce additional terms.

References

Hunter, The Heat and Schrödinger Equations, §§5.1–5.3.

Uniqueness from terminal data

Backward uniqueness holds for the L2 heat evolution even though arbitrary terminal data do not yield a well-posed backward problem.