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Mathematics LibreTexts

6: Parabolic Equations

Here we consider linear parabolic equations of second order. An example is the heat equation

$$u_t=a^2\triangle u,$$

where \(u=u(x,t)\), \($\in\mathbb{R}^3\), \(t\ge0\), and \(a^2\) is a positive constant called conductivity coefficient. The heat equation has its origin in physics where \(u(x,t)\) is the temperature at \(x\) at time \(t\), see [20], p. 394, for instance.

Remark 1. After scaling of axis we can assume \(a=1\).

Remark 2. By setting \(t:=-t\), the heat equation changes to an equation which is called backward equation. This is the reason for the fact that  the heat equation describes irreversible processes in contrast to the wave equation \(\Box u=0\) which is invariant with respect the mapping \(t\mapsto -t\). Mathematically, it means that it is not possible, in general, to find the distribution of temperature at an earlier time \(t<t_0\) if the distribution is given at \(t_0\).

Consider the initial value problem for \(u=u(x,t)\), \(x\in\mathbb{R}^n\), \(t\ge0\) and \(u\in C^\infty(\mathbb{R}^n\times R_+)\),
u_t&=&\triangle u\\

where \(\phi\in C(\mathbb{R}^n)\) is given and \(\triangle\equiv\triangle_x\).

Thumbnail: A visualization of a solution to the two-dimensional heat equation with temperature represented by the third dimension. Imaged used wth permission (Public Domain; Oleg Alexandrov). The heat equation is a parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time.