The Klein-Gordon equation (scaling constants) is
$$\square u = -m^2 u.$$
I am wondering if the equation
$$\square u = m^2 u.$$
for real $m$ ever shows up in the physical literature?
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I believe this is one equation used to study Tachyons which would have a complex mass.
In the normal equation the mass term is (of course) $m^2$, but in your modified form the mass term would be $-m^2$. The mass of a tachyon would be $im$ in your case. One example of this is in Revisiting Barry Cox and James Hill’s theory of superluminal motion: a possible solution to the problem of spinless tachyon localization in section 4.