I've just started reading Partial Differential Equations by Lawrence Evans, and I'm already confused by the notation on page 2. The author defines the following classes of PDEs:
DEFINITIONS.
(i) The partial differential equation (1) is called linear if it has the form $$ \sum_{|\alpha| \leq k} a_{\alpha}(x)D^{\alpha} u = f(x) $$ for given functions $a_{\alpha} \; (|\alpha| \leq k), f$. This linear PDE is homogeneous if $f \equiv 0$.(ii) The PDE (1) is semilinear if it has the the form $$ \sum_{|\alpha| = k} a_{\alpha}(x)D^{\alpha} u + a_0(D^{k-1} u, \ldots, Du,u,x) = 0. $$
(iii) The PDE (1) is quasilinear if it has the form $$ \sum_{|\alpha| = k} a_{\alpha}(D^{k-1}u,\ldots,Du,u,x) D^{\alpha} u + a_0(D^{k-1}u,\ldots,Du,u,x) = 0. $$
(iv) The PDE (1) is fully nonlinear if it depends nonlinearly upon the highest order derivatives.
My questions: Where do time derivatives fit into the above classifications? It seems to me that the above definitions concern PDEs which involve $u$ and its spatial derivatives only. This idea seems to be reinforced on the following page where Evans writes:
Throughout $x \in U$, where $U$ is an open subset of $\mathbb{R}^n$, and $t \geq 0$. Also, $Du = D_x u = (u_{x_1},\ldots,u_{x_n})$ denotes the gradient of $u$ with respect to the spatial variable $x = (x_1,\ldots,x_n)$. The variable $t$ always denotes time.
So it appears that the $D^k$'s denote spatial derivatives only. But I think this must be wrong; "most" PDEs involve time derivatives. For example, how would the PDE $$ u_t + \sum_{i=1}^{n} b^i u_{x_i} = 0 $$ fit into the above classification, given the $u_t$ term?