I am doing a research using PDE which is a little different from the standard Parabolic type. The following is my case:
\begin{equation} Lu = - \sum_{i, j = 1}^{N}a^{i,j}(x, t)u_{x_{i}x_{j}} + \sum_{k = 1}^{N}b^{i}(x, t)u_{x_{i}} + c(x, t)u \end{equation} where $A = (a^{i,j})_{i, j = 1}^{N}$ is a Positive Semi-definite symmetric matrix and $c(x, t) = c$ is constant. Furthermore, every row and column of $A$ has sum of zero and non-negative diagonal terms.
According to the Evans' book, when there is a constant $\theta > 0$ such that
\begin{equation} \sum_{k = 1}^{N}a^{i,j}(x, t)\psi_{i}\psi_{j} \geq \theta|\psi|^{2} \end{equation} for all $(x, t) \in U_{T}$ and $\psi \in \mathbf{R}^{N}$, we say that the partial differential operator $\frac{\partial}{\partial t} + L$ is (uniformly) parabolic. But in my case, with $\psi = (1, 1,...,1)$, $\sum_{k = 1}^{N}a^{i,j}(x, t)\psi_{i}\psi_{j} = 0$ because every row and column of $A$ has sum of zero. This implies my PDE is not a Parabolic type.
With my PDE above, I would like to show the existence and uniqueness of solution. It would be highly appreciated if you recommend me any references about this stuff. Thanks.