Properties of Klein-Gordon solutions

Question

Consider the differential equation in $\mathbb{R}^d$: $$ (\Box+m^2)\psi(\vec{r})=0 $$ Does $\psi$ have any special properties such as harmonic functions? For example, does it determined over an open region $\Omega$ only by its values on $\partial\Omega$? Does it satisfy the mean value property? In what senses, the properties of solutions to the Klein-Gordon equation differ from harmonic functions?

Answer 1

Solution of such equation is determined by value of the function and its first time derivative at initial time. There aren't many similarities with Laplace equation, basically because Klein Gordon differential operator is hyperbolic, while Laplace operator is elliptic. These are the buzz words you should look at.