Section 6.1 Linear parabolic operators
In this section and the next we consider linear operators of the form
\begin{equation}
L = -\partial_t + a_{ij}(x,t) \partial_{ij} + b_i(x,t) \partial_i + c(x,t)\tag{6.1}
\end{equation}
for \((x,t) \in \Omega\times(0,T)\) where \(\Omega\) an open subset of \(\R^N\text{.}\) We think of \(x\) as a spatial variable and \(t\) as time. Recall from Notation 2.12 that
\begin{gather*}
\partial_i = \frac\partial{\partial x_i},
\qquad
\partial_{ij} = \frac{\partial^2}{\partial x_i\, \partial x_j},
\end{gather*}
are partial derivatives with respect to the spatial variable \(x\) variable only. Thus the implicit sums in (6.1) range over \(i,j = 1,\ldots,N\text{.}\) Figure 6.1 illustrates several space-time sets which we will use,
\begin{align*}
D\amp= \Omega\times(0,T],\\
Q\amp= \Omega\times(0,T),\\
\Sigma\amp= (\partial \Omega\times[0,T]) \cup (\Omega\times\{0\}).
\end{align*}
The set \(D\) is sometimes called the parabolic interior, while \(\Sigma\) is called the parabolic boundary.
We shall always assume that the coefficient functions \(a_{ij},b_i,c\) are bounded \(\overline D\) and that \(a_{ij}=a_{ji}\text{.}\) We call \(L\) in (6.1) uniformly parabolic if
\begin{gather}
a_{ij}(x,t)\xi_i \xi_j \ge \lambda_0 |\xi|^2\tag{6.2}
\end{gather}
for some constant \(\lambda_0 \gt 0\) independent of \(\xi \in \R^N\) and \((x,t) \in \overline
D\text{.}\)
Remark 6.1.
Just as the prototypical uniformly elliptic operator is the Laplacian \(\Delta\text{,}\) the prototypical uniformly parabolic operator is the heat operator \(\partial_t - \Delta\text{.}\)
Remark 6.2.
In applications, the parabolic equation \(Lu=f\) is usually supplemented by boundary conditions for \(u\) on the parabolic boundary \(\Sigma\) — initial data at \(t=0\) together with boundary data on \(\partial\Omega\) for all relevant \(t\) — and one attempts to solve for the values of \(u\) in the parabolic interior \(D\text{.}\)
