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does the equation $ u_{tt} + \Delta u = f$ makes any sense?

The usual wave equation is $ u_{tt} - c^2\Delta u = f$ What would happen if we changed the sign to the Laplace operator? Physically speaking, would it still represent anything?

Matt
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    I think this is a deeper question than you may expect. There is an industry of sorts built out of taking established evolution PDEs and then treating them in imaginary time. For instance the (nonrelativistic) Schrodinger equation in the absence of a potential can be viewed as the heat equation in imaginary time. The wave equation with the sign flipped can be viewed as the wave equation with imaginary velocity, or equivalently in imaginary time. – Ian Jun 03 '15 at 16:23
  • Many disciplines assign various meaning to complex numbers. You could interpret a complex number as the phase between two waves, as a vector in two dimensions... So it would be possible to construct (if it does not already exist) a frame work that will have a physically tangible meaning for a complex speed. – Gappy Hilmore Jun 03 '15 at 16:27
  • Another way to view this is to transform into the frequency domain. The "usual" wave equation transforms to the Helmholtz equation $\nabla^2 \hat u(\vec r,\omega)+k^2\hat u(\vec r,\omega)=\hat f(\vec r,\omega)$, where $k=\omega^2/c^2$ and the hat symbol designates the Fourier transform. The equation of question transforms to the equation $\nabla^2 \hat u(\vec r,\omega)-\omega^2\hat u(\vec r,\omega)=\hat f(\vec r,\omega)$. The basic difference is a $90$ degree rotation of the inverse transform complex plane. – Mark Viola Jun 03 '15 at 16:35

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The left-hand side will then simply be the Laplace operator in the variables $(t,x_1,x_2,...)$, so what you get is the Poisson equation.

Hans Lundmark
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  • Technically, this is correct, but the character of the Laplace/Poisson equation when formulated as an IVP is rather different from its character when formulated as a BVP or on the full space. – Ian Jun 03 '15 at 21:46
  • Yes, of course, but the question doesn't mention anything about boundary or initial conditions. – Hans Lundmark Jun 04 '15 at 06:15
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    Not explicitly, but it calls the additional variable $t$, thinks of the equation as "the wave equation with the sign flipped", and asks for a physical interpretation. That clearly suggests an IVP in time is intended. – Ian Jun 04 '15 at 14:15
  • So you have an elliptic problem in 4-dim Euclidean space instead of a hyperbolic one.. in Physics, elliptic equations are well suited to describe equilibrium states rather than propagating ones (hyperbolic). For elliptic PDE, see e.g. https://math.stackexchange.com/q/586837/532409 , https://math.stackexchange.com/q/857899/532409 – Quillo Jun 08 '22 at 08:30