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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?

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  • $\begingroup$ Evans talks about parabolic PDEs in the middle section of the book. $\endgroup$
    – Mr. Brown
    Commented Jul 19, 2023 at 18:05
  • $\begingroup$ I mean you can think of one of the $x_i's$ as the time dimension if you want. Keeping $t$ and $\vec{x}$ separately is a notational convenience. $\endgroup$
    – dezdichado
    Commented Jul 19, 2023 at 18:06

1 Answer 1

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The classification into linear/nonlinear PDE is independent of whether we view one of the variables as time or not. Your second quote is from section 1.2 (Examples), and the notation introduced there applies to this section, not the previous section.

The PDE $$ u_t + \sum_{i=1}^{n} b^i u_{x_i} = 0 \tag{1} $$ is linear in each term, so it is a linear PDE. In fact, it is given as an example in section 1.2.1. a. (Linear equations).

As you can see from the rest of the examples, it is rare for a PDE to depend nonlinearly on a time derivative. We usually interperet a variable $t$ as time if the PDE is of the form $\partial_t u = L(u)$ or $\partial_t^2 u + d \partial_t u = L(u)$ where $L$ is a (possibly nonlinear) partial differential operator involving derivatives of the remaining variables (which are then intepreted as spatial).

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    $\begingroup$ Thanks for your answer, this is helpful. So basically, in the examples in section 1.2 we think of $t$ as $x_{n+1}$ when categorizing it according to the given definitions of linear, semilinear, etc.? $\endgroup$
    – Leonidas
    Commented Jul 19, 2023 at 18:35

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