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For the following wave equation

$ \frac{{\partial ^2 p}}{{\partial ^2 x}} + \frac{{\partial ^2 p}}{{\partial ^2 y}} = A\frac{{\partial ^2 p}}{{\partial ^2 t}} + B\frac{{\partial p}}{{\partial t}} $

is there a way to show that there are boundary conditions at or near positive and negative infinity, for both non-zero B and B=0 conditions, and for {A,B} as rational numbers? I believe that this should follow from Sommerfeld's condition of radiation, and should perhaps be similar to conditions for the ordinary wave equation. What are these boundary conditions? Ideally, I think that the boundary conditions should involve both time and spatial derivatives.

By "positive and negative infinity" I mean that I am interested in what happens when $x \to \pm \infty $ and $y \to \pm \infty$. I've been working on a problem where I would like to computationally solve the wave equation with boundary conditions that approximate infinity. So I suppose that this would be an imposed compatibility condition.

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  • $\begingroup$ What infinity? Do you mean time-like infinity or null infinity? (Are you looking at infinity in the $t\to \pm\infty$ direction, or $r \pm t\to \pm \infty$ direction?) I also don't understand what you mean by boundary conditions. Do you mean scattering data (as in for each classical solution there is a unique function defined on the null boundary $\mathcal{I}^\pm$ which captures the whole function) or do you mean boundary condition as some sort of imposed compatibility condition to restrict the set of solutions for a PDE? $\endgroup$
    – Willie Wong
    Jul 24, 2010 at 9:21
  • $\begingroup$ Just a guess what is meant by "boundary conditions": There is the classical problem of uniqueness of solutions of PDEs. And often to have uniqueness, one needs to impose some kind of boundary conditions. Of course it is not clear here, what problem is being solved. E.g. are thee initial conditions, if yes where, and so on. But I guess this is part of the question, what kind of conditions do we need to pose the question? If this is so, the original poster will need to supply more information, because right now, I can guess many answers, most probably not helpful. $\endgroup$
    – Helge
    Jul 24, 2010 at 10:29
  • $\begingroup$ Also, when $B\neq 0$, the term introduced is a friction term. So the energy of the solution will tend to be exponential in time. In fact, standard energy estimates should tell you that when $B > 0$ the solution decays exponentially as $t\to\infty$, but it can potentially grow very fast as $t\to -\infty$. $\endgroup$
    – Willie Wong
    Jul 24, 2010 at 11:48

1 Answer 1

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First of all, can you be more precise in your question? You are asking about boundary conditions at infinity, and this might make sense, but... for what purpose? do you need a set of conditions that imply existence and uniqueness of a global solution? or, do you need to classify solutions of the standard Cauchy problem (with data at t=0) according to their behaviour at infinity?

Anyway, there are a couple general tools that might help you at least to clarify what you are looking for exactly:

1) If you need a tool to classify solutions according to their behaviour at infinity, then scattering theory (mentioned by Willie in his comment) might be helpful. However, its main purpose is to compare two different equations, i.e., use the solutions of a simpler equation to classify the solutions of a 'more difficult' equation. So I do not think this is what you actually need.

2) If you need to understand what might be reasonable 'data at infinity' for a Cauchy problem, then the Kelvin transform might be of use. This is a space-time change of coordinates that transforms a wave equation into a wave equation, and exchanges infinity with t=0. Playing with it might give you some insight into what kind of conditions you might impose at infinity on your solution. There is also a much more sofisticated transform with a similar effect, the Penrose transform, but this might be overkill in your case.

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  • $\begingroup$ Sure, number (2) is exactly what I am looking for; so thank you! Could you clarify what is meant by "infinity" as per Willie's comment above? Could you suggest a reference (book/paper/monograph) dealing with application of Kelvin transforms? $\endgroup$
    – Nicholas Kinar
    Jul 24, 2010 at 19:35
  • $\begingroup$ I think the hyperbolic version of the Kelvin transform was introduced by K.Morawetz, so this is old, but not as old as Lord Kelvin himself. A starting point could be the paper jstor.org/pss/2154859 where it is put to good use for problems of global existence of nonlinear wave equations. $\endgroup$
    – Piero D'Ancona
    Jul 24, 2010 at 21:54
  • $\begingroup$ @Piero: interesting paper. I haven't looked at it too closely, but I assume the restriction to odd dimensions is related to the strong Huygen's principle? The fact that the data has support restricted to the wave-zone and the construction of the approximate solutions seems to me to be a sort of geometric optics technique. It is quite interesting and different from my usual mode of thinking. $\endgroup$
    – Willie Wong
    Jul 25, 2010 at 11:19
  • $\begingroup$ @Willie: I agree that the restriction to odd dimensions is indeed interesting. Thanks for pointing this out. $\endgroup$
    – Nicholas Kinar
    Jul 25, 2010 at 15:21
  • $\begingroup$ @Willie: my feeling is that the requirement N odd is just a technical one. Having the strong Huygens principles available makes for easier equations to solve when constructing the expansion, but I guess it should not be essential for the result to hold. $\endgroup$
    – Piero D'Ancona
    Jul 26, 2010 at 12:29

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