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I am interested in solutions to the linear Cauchy problem $$\Bigl(\frac{\partial^2}{\partial t^2} + a(t, x)\frac{\partial}{\partial t} + \sum_{j=1}^n b_j(t, x) \frac{\partial}{\partial x_j}\Bigr)u(t, x) = h(t, x)$$ with initial data $$u(0, x) = u_0(x), ~~~~~~~~~ \frac{\partial}{\partial t}u(0, x) - u_1(x).$$ Of course, in the case that everything is analytic, there is a unique analytic solution by Cauchy-Kowalewskaja, but I canot make this assumption.

If (hopefully), there is a unique solution for (at least some) given data, I would also be interested in growth estimates on the solution in $t$ (in fact, I do have that the $b_j$ have exponential decay in $t$).

Any hints and references on how to tackle this problem will be appreciated!

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  • $\begingroup$ This is like solving the heat equation "sideways," which is well known not to be a well posed problem. $\endgroup$
    – Michael Renardy
    Jun 19, 2014 at 20:52
  • $\begingroup$ In what way would that be the same? What exactly do you mean by solving the heat equation sideways? $\endgroup$
    – Matthias Ludewig
    Jun 20, 2014 at 8:26
  • $\begingroup$ Take the standard heat equation in 1 dimension, interchange $t$ and $x$, you get something that looks like your equation. It does not have continuous dependence on initial data: see e.g. Problem 15 of this set of notes. $\endgroup$
    – Willie Wong
    Jun 20, 2014 at 8:40

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