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In Leveque's "Numerical Methods for Conservation Laws", Ch. 3.1.1., he says that given a system of Hyperbolic PDE's and a point $(\bar{t},\bar{x} )$, its domain of dependence $D(\bar{t},\bar{x} )$ is always a bounded set. The domains of dependence here is defined by the minimal size of an initial data set needed to determine the solution's value at a given point.

My question: Wikipedia defines hyperbolicity of a system by its eigenvalues. How can I proceed from this definition to prove the boundedness of the domain of dependence?

Thanks

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    $\begingroup$ (1) Leveque's use of the word "domain of dependence" is non-standard, usually one refers to the entire space-time domain and not just its intersection with the initial data hypersurface. (2) This result is old and standard (going back to at least Leray), for strictly or symmetric hyperbolic systems; look for example in Volume 2 of Courant-Hilbert where they discuss hyperbolic PDEs. (3) The property you are referring to is also called the "finite speed of propagation" property for hyperbolic PDEs, and is proven in most textbooks in hyperbolic PDEs. $\endgroup$
    – Willie Wong
    Commented Sep 2, 2016 at 14:53
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    $\begingroup$ For the sake of giving a reasonably complete reference: the case of strictly hyperbolic systems with variable coefficients is treated by Jean Leray in his notes on Hyperbolic Differential Equations; for first order hyperbolic systems, you can see this article by Rauch and his references. $\endgroup$
    – Willie Wong
    Commented Sep 2, 2016 at 15:25
  • $\begingroup$ As Willie mentioned, the domain of dependence theorem is discussed in many textbooks and monographs (see the answers to MO181630 for a sample of references). Of course, there are many (slightly distinct) notions of hyperbolicity and the proof needs to be adapted to each case. $\endgroup$
    – Igor Khavkine
    Commented Sep 3, 2016 at 7:15
  • $\begingroup$ @WillieWong, thanks, here's a full link to Rauch's paper - projecteuclid.org/download/pdf_1/euclid.maa/1175797445 $\endgroup$
    – Amir Sagiv
    Commented Sep 3, 2016 at 7:50
  • $\begingroup$ @IgorKhavkine: do you know which of the items in your list deal with the case of higher order hyperbolic systems? (Leray is the only one I checked since it was the only one I had in my office; I know also that several on your list only treat first/second order equations.) $\endgroup$
    – Willie Wong
    Commented Sep 3, 2016 at 14:55

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