In this paper, we consider wave equations with double damping terms expressed by $$u_{t}$$ ut and $$-\Delta u_{t}$$ -Δut and a power type of nonlinearity $$\vert u\vert ^{p}$$ |u|p . We are concerned with mixed problems for these equations in exterior domains of a bounded obstacle. A main purpose is to determine a so-called critical exponent of the power p of the nonlinearity $$\vert u\vert ^{p}$$ |u|p . In particular, in the two dimensional case, our results are optimal, and the critical exponent is given by the Fujita one. This shows a parabolic aspect (as $$t \rightarrow \infty $$ t→∞ ) of our equations considered in exterior domains, and one can see that the usual frictional damping $$u_{t}$$ ut is more dominant than the strong one $$-\Delta u_{t}$$ -Δut as $$t \rightarrow \infty $$ t→∞ even in the nonlinear problem case.