About the wave equation

Question

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?

Answer 1

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.