a question about first-order hyperbolic equations

Question

Performing certain manipulations on pseudo-differential equations I have come across the following first order equation: $$ D_{t}u-\lambda(t,x,D_{t},D_{x})u=0, \ \ (*) $$ where $\lambda$ is a scalar pseudo-differential operator with the principal symbol being real-valued and independent of $D_{t}.$ But, the lower-order terms of $\lambda$ depend on $D_{t}$ (or $\tau$ at the symbol level).

I was expecting a hyperbolic equation. But, I find that standard text books(like M.Taylor's 'Pseudo-differential operators') treat only equations in which $\lambda$ term is independent of $D_{t}$ (or $\tau$ at the symbol level).

For the equation $(*)$ to be hyperbolic, is it necessary that $\lambda$ to be independent of $D_{t}$? Are there any references which discuss these issues?

Answer 1

Your principal symbol is $\tau-\lambda_1 (t,x,\xi)$, with a real-valued $\lambda_1$, but you have lower order terms which may depend on $\tau$. You can get rid of that dependency as follows: take for instance $\lambda_0$ of order $0$. You can find an operator $M$ of order $0$ such that $$ Op\bigl(\tau-\lambda_1 (t,x,\xi)-\lambda_0(t,x,\tau,\xi)\bigr)= e^{-iM}Op\bigl(\tau-\lambda_1 (t,x,\xi)\bigr)e^{iM}+Op(S^{-1}). $$ In fact, since $e^{iM}$ is a pseudodifferential operator of order 0, the composition formula gives $$ e^{-iM}Op\bigl(\tau-\lambda_1 (t,x,\xi)\bigr)e^{iM}=D_t+\frac{\partial M}{\partial t} -e^{-iM}[Op(\lambda_1),e^{iM}]-Op(\lambda_1), $$ and since $ e^{-iM}[Op(\lambda_1),e^{iM}]=Op(${$\lambda_1,m$}$)+Op(S^{-1}) $ where $m$ is of order 0 and {} is the Poisson bracket. To get rid of $\lambda_0$, you have only to solve $$ \frac{\partial m}{\partial t}-\text{ {$\lambda_1,m$ } }=-\lambda_0, $$ which is a linear transport equation of real principal type in $m$. You can of course iterate this business to get a remainder of order as negative as you like. This is explained in the Chapter 23 on hyperbolic equations in Hörmander third volume of ALPDO.

Answer 2

If I understand correctly, you have a PDE of the form $$ (D_t - \lambda_1(t, x, D_x) + \lambda_0(t, x, D_t, D_x))u = f, $$ where $\lambda_1$ is a first order pseudodifferential operator and $\lambda_0$ is a zero-th order pseudodifferential operator. It seems to me that the proofs of many if not all estimates, including energy integral estimates, and theorems about regularity, uniqueness, and existence for the equation $$ (D_t - a(t,x)D_x + b(t,x))u = f, $$ as presented in books like Taylor can be extended to your PDE.