Parabolic partial differential equation

Definition

Let $U \subseteq \R^n$ be open, let $T > 0$, and write the variables in $U \times (0, T)$ as $(x, t)$. Consider a linear PDE in $U \times (0, T)$ of the following form. $$\partial_t u - \sum_{i,j=1}^{n} a_{ij}(x, t) \, \partial_i \partial_j u + \sum_{i=1}^{n} b_i(x, t) \, \partial_i u + c(x, t) \, u = f(x, t).$$ Here all coefficients are real, $a_{ij} = a_{ji}$, and $\partial_i$ is the derivative in $x_i$. The equation is a second-order linear parabolic equation if its spatial part is uniformly elliptic. That is, there is $\theta > 0$ with $$\sum_{i,j=1}^{n} a_{ij}(x, t) \, \xi_i \xi_j \ge \theta \abs{\xi}^2 \quad \text{for all } (x, t) \in U \times (0, T), \ \xi \in \R^n.$$