PARABOLIC EQUATIONS AND THE ADVECTION-DIFFUSION EQUATION

The simplest example is the ``Heat Equation''
Let $U=U(x,t)$, and $t>0$. The Heat Equation is

$$\left\{\begin{array}{ll} U_t=bU_{xx} & b>0\quad \mbox{ real, called \lq\lq dissipation constant''}\\ U(0,x)+U_0(x) & \quad \end{array}\right.$$

plus boundary values in $x$. It is a boundary-initial value problem, but for now, take $x\in {\mathbb{R}}^1$.