Lifespan estimates for critical semilinear wave equations and scale invariant damped wave equations in exterior domain in high dimensions

In this paper, we consider semilinear wave equations with critical power and scale invariant damping term 2(1 + t)(-1)u(t) in exterior domain in high dimensions (n >= 3). Upper bound lifespan estimates of solution are established by employing test function method. The novelty is that we show the asymptotic behavior of the test function by using maximum principle. Comparing with the method utilized in Sobajima (J Math Anal Appl 484:123667, 2019), we avoid using the modified Bessel functions to construct the test function. It is worth to mention that the method employed in this paper is also different from the one in Lai (Nonlinear Anal 125: 550-560, 2015) and Lai (Nonlinear Anal. 143: 89-104, 2016).